Artin transfer (group theory)#Homomorphisms

Let $G$ be a group and $H\backslash le\; G$ be a subgroup of finite index $n.$

Definitions.{{cite book|author=Huppert, B.|year=1979|title=Endliche Gruppen I|publisher=Grundlehren der mathematischen Wissenschaften, Vol. 134, Springer-Verlag Berlin Heidelberg New York}} A left transversal of $H$ in $G$ is an ordered system $(g\_1,\backslash ldots,g\_n)$ of representatives for the left cosets of $H$ in $G$ such that

:$G=\backslash bigsqcup\_\{i=1\}^n\; g\_iH.$

Similarly a right transversal of $H$ in $G$ is an ordered system $(d\_1,\; \backslash ldots,\; d\_n)$ of representatives for the right cosets of $H$ in $G$ such that

:$G=\backslash bigsqcup\_\{i=1\}^n\; Hd\_i.$

Remark. For any transversal of $H$ in $G$, there exists a unique subscript $1\backslash le\; i\_0\backslash le\; n$ such that $g\_\{i\_0\}\backslash in\; H$, resp. $d\_\{i\_0\}\backslash in\; H$. Of course, this element with subscript $i\_0$ which represents the principal coset (i.e., the subgroup $H$ itself) may be, but need not be, replaced by the neutral element $1$.

Lemma. Let $G$ be a non-abelian group with subgroup $H$. Then the inverse elements $(g\_1^\{-1\},\backslash ldots,g\_n^\{-1\})$ of a left transversal $(g\_1,\backslash ldots,g\_n)$ of $H$ in $G$ form a right transversal of $H$ in $G$. Moreover, if $H$ is a normal subgroup of $G$, then any left transversal is also a right transversal of $H$ in $G$.

:Proof. Since the mapping $x\backslash mapsto\; x^\{-1\}$ is an involution of $G$ we see that:

::$G=G^\{-1\}=\backslash bigsqcup\_\{i=1\}^n\; (g\_iH)^\{-1\}=\backslash bigsqcup\_\{i=1\}^n\; H^\{-1\}g\_i^\{-1\}=\backslash bigsqcup\_\{i=1\}^n\; Hg\_i^\{-1\}.$

:For a normal subgroup $H$ we have $xH=Hx$ for each $x\backslash in\; G$.

We must check when the image of a transversal under a homomorphism is also a transversal.

Proposition. Let $\backslash phi:\; G\backslash to\; K$ be a group homomorphism and $(g\_1,\backslash ldots,g\_n)$ be a left transversal of a subgroup $H$ in $G$ with finite index $n.$ The following two conditions are equivalent:

• $(\backslash phi(g\_1),\backslash ldots,\backslash phi(g\_n))$ is a left transversal of the subgroup $\backslash phi(H)$ in the image $\backslash phi(G)$ with finite index $(\backslash phi(G):\backslash phi(H))=n.$
• $\backslash ker(\backslash phi)\backslash le\; H.$

:Proof. As a mapping of sets $\backslash phi$ maps the union to another union:

::$\backslash phi(G)=\backslash phi\; \backslash left(\backslash bigcup\_\{i=1\}^n\; g\_iH\; \backslash right\; )=\backslash bigcup\_\{i=1\}^n\; \backslash phi(g\_iH)=\backslash bigcup\_\{i=1\}^n\; \backslash phi(g\_i)\backslash phi(H),$

:but weakens the equality for the intersection to a trivial inclusion:

::$\backslash emptyset=\backslash phi(\backslash emptyset)=\backslash phi(g\_iH\backslash cap\; g\_jH)\backslash subseteq\backslash phi(g\_iH)\backslash cap\backslash phi(g\_jH)=\backslash phi(g\_i)\backslash phi(H)\backslash cap\backslash phi(g\_j)\backslash phi(H),\; \backslash qquad\; i\; \backslash ne\; j.$

:Suppose for some $1\backslash le\; i\backslash le\; j\backslash le\; n$:

::$\backslash phi(g\_i)\backslash phi(H)\backslash cap\backslash phi(g\_j)\backslash phi(H)\backslash ne\backslash emptyset$

:then there exists elements $h\_i,h\_j\backslash in\; H$ such that

::$\backslash phi(g\_i)\backslash phi(h\_i)=\backslash phi(g\_j)\backslash phi(h\_j)$

:Then we have:

::$\backslash begin\{align\}$

\phi(g_i)\phi(h_i)=\phi(g_j)\phi(h_j) &\Longrightarrow \phi(g_j)^{-1}\phi(g_i)\phi(h_i)\phi(h_j)^{-1}=1 \\

&\Longrightarrow \phi \left (g_j^{-1}g_ih_ih_j^{-1} \right )= 1 \\

&\Longrightarrow g_j^{-1}g_ih_ih_j^{-1}\in\ker(\phi) \\

&\Longrightarrow g_j^{-1}g_ih_ih_j^{-1}\in H && \ker(\phi)\le H \\

&\Longrightarrow g_j^{-1}g_i \in H && h_ih_j^{-1}\in H \\

&\Longrightarrow g_iH=g_jH \\

&\Longrightarrow i = j

\end{align}

:Conversely if $\backslash ker(\backslash phi)\; \backslash nsubseteq\; H$ then there exists $x\backslash in\; G\backslash setminus\; H$ such that $\backslash phi(x)=1.$ But the homomorphism $\backslash phi$ maps the disjoint cosets $x\backslash cdot\; H\backslash cap\; 1\; \backslash cdot\; H=\backslash emptyset$ to equal cosets:

::$\backslash phi(x)\backslash phi(H)\backslash cap\backslash phi(1)\backslash phi(H)=1\backslash cdot\backslash phi(H)\backslash cap\; 1\backslash cdot\backslash phi(H)=\backslash phi(H).$

Remark. We emphasize the important equivalence of the proposition in a formula:

:$(1)\backslash quad\; \backslash ker(\backslash phi)\backslash le\; H\; \backslash quad\; \backslash Longleftrightarrow\backslash quad\; \backslash begin\{cases\}\; \backslash phi(G)=\backslash bigsqcup\_\{i=1\}^n\; \backslash phi(g\_i)\backslash phi(H)\; \backslash \backslash (\backslash phi(G):\backslash phi(H))=n\; \backslash end\{cases\}$

Suppose $(g\_1,\backslash ldots,g\_n)$ is a left transversal of a subgroup $H$ of finite index $n$ in a group $G$. A fixed element $x\backslash in\; G$ gives rise to a unique permutation $\backslash pi\_x\backslash in\; S\_n$ of the left cosets of $H$ in $G$ by left multiplication such that:

:$(2)\backslash quad\; \backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; xg\_iH=g\_\{\backslash pi\_x(i)\}H\; \backslash Longrightarrow\; xg\_i\backslash in\; g\_\{\backslash pi\_x(i)\}H.$

Using this we define a set of elements called the monomials associated with $x$ with respect to $(g\_1,\backslash ldots,g\_n)$:

:$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; u\_x(i):=g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i\backslash in\; H.$

Similarly, if $(d\_1,\backslash ldots,d\_n)$ is a right transversal of $H$ in $G$, then a fixed element $x\backslash in\; G$ gives rise to a unique permutation $\backslash rho\_x\backslash in\; S\_n$ of the right cosets of $H$ in $G$ by right multiplication such that:

:$(3)\backslash quad\; \backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; Hd\_ix=Hd\_\{\backslash rho\_x(i)\}\; \backslash Longrightarrow\; d\_ix\backslash in\; Hd\_\{\backslash rho\_x(i)\}.$

And we define the monomials associated with $x$ with respect to $(d\_1,\backslash ldots,d\_n)$:

:$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; w\_x(i):=d\_ixd\_\{\backslash rho\_x(i)\}^\{-1\}\backslash in\; H.$

Definition. The mappings:

:$\backslash begin\{cases\}\; G\backslash to\; S\_n\; \backslash \backslash \; x\backslash mapsto\backslash pi\_x\; \backslash end\{cases\}\; \backslash qquad\; \backslash begin\{cases\}\; G\backslash to\; S\_n\; \backslash \backslash \; x\backslash mapsto\backslash rho\_x\; \backslash end\{cases\}$

are called the permutation representation of $G$ in the symmetric group $S\_n$ with respect to $(g\_1,\backslash ldots,g\_n)$ and $(d\_1,\backslash ldots,d\_n)$ respectively.

Definition. The mappings:

:$\backslash begin\{cases\}\; G\backslash to\; H^n\backslash times\; S\_n\; \backslash \backslash \; x\backslash mapsto(u\_x(1),\backslash ldots,u\_x(n);\backslash pi\_x)\backslash end\{cases\}\; \backslash qquad\; \backslash begin\{cases\}\; G\backslash to\; H^n\backslash times\; S\_n\; \backslash \backslash \; x\backslash mapsto(w\_x(1),\backslash ldots,w\_x(n);\backslash rho\_x)\; \backslash end\{cases\}$

are called the monomial representation of $G$ in $H^n\backslash times\; S\_n$ with respect to $(g\_1,\backslash ldots,g\_n)$ and $(d\_1,\backslash ldots,d\_n)$ respectively.

Lemma. For the right transversal $(g\_1^\{-1\},\backslash ldots,g\_n^\{-1\})$ associated to the left transversal $(g\_1,\; \backslash ldots,\; g\_n)$, we have the following relations between the monomials and permutations corresponding to an element $x\backslash in\; G$:

:

:Proof. For the right transversal $(g\_1^\{-1\},\backslash ldots,g\_n^\{-1\})$, we have $w\_x(i)=g\_i^\{-1\}xg\_\{\backslash rho\_x(i)\}$, for each $1\backslash le\; i\backslash le\; n$. On the other hand, for the left transversal $(g\_1,\backslash ldots,g\_n)$, we have

::$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; u\_x(i)^\{-1\}=\; \backslash left\; (g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i\; \backslash right\; )^\{-1\}=g\_i^\{-1\}x^\{-1\}g\_\{\backslash pi\_x(i)\}=g\_i^\{-1\}x^\{-1\}g\_\{\backslash rho\_\{x^\{-1\}\}(i)\}=w\_\{x^\{-1\}\}(i).$

:This relation simultaneously shows that, for any $x\backslash in\; G$, the permutation representations and the associated monomials are connected by $\backslash rho\_\{x^\{-1\}\}=\backslash pi\_x$ and $w\_\{x^\{-1\}\}(i)=u\_x(i)^\{-1\}$ for each $1\backslash le\; i\backslash le\; n$.

Definitions.{{cite journal|author=Schur, I.|year=1902| title=Neuer Beweis eines Satzes über endliche Gruppen| journal= Sitzungsb. Preuss. Akad. Wiss.|pages=1013–1019}}{{cite journal|author=Artin, E.|year=1929| title= Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz|journal=Abh. Math. Sem. Univ. Hamburg|volume=7|pages=46–51|doi=10.1007/BF02941159|s2cid=121475651}} Let $G$ be a group and $H$ a subgroup of finite index $n.$ Assume $(g)\; =\; (g\_1,\backslash ldots,g\_n)$ is a left transversal of $H$ in $G$ with associated permutation representation $\backslash pi\_x:\; G\backslash to\; S\_n,$ such that

:$\backslash forall\; i\backslash in\backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; u\_x(i)\; :=g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i\backslash in\; H.$

Similarly let $(d)\; =\; (d\_1,\backslash ldots,d\_n)$ be a right transversal of $H$ in $G$ with associated permutation representation $\backslash rho\_x:\; G\; \backslash to\; S\_n$ such that

:$\backslash forall\; i\backslash in\backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; w\_x(i):=d\_ixd\_\{\backslash rho\_x(i)\}^\{-1\}\backslash in\; H.$

The Artin transfer $T\_\{G,H\}^\{(g)\}:\; G\backslash to\; H/H\text{'}$ with respect to $(g\_1,\backslash ldots,g\_n)$ is defined as:

:$(5)\backslash quad\; \backslash forall\; x\; \backslash in\; G:\; \backslash qquad\; T\_\{G,H\}^\{(g)\}(x)\; :=\; \backslash prod\_\{i=1\}^n\; g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i\backslash cdot\; H\text{'}\; =\; \backslash prod\_\{i=1\}^n\; u\_x(i)\backslash cdot\; H\text{'}.$

Similarly we define:

:$(6)\backslash quad\; \backslash forall\; x\; \backslash in\; G:\; \backslash qquad\; T\_\{G,H\}^\{(d)\}(x)\; :=\; \backslash prod\_\{i=1\}^n\; d\_ixd\_\{\backslash rho\_x(i)\}^\{-1\}\backslash cdot\; H\text{'}\; =\backslash prod\_\{i=1\}^n\; w\_x(i)\backslash cdot\; H\text{'}.$

Remarks. Isaacs calls the mappings

:$\backslash begin\{cases\}\; P:\; G\backslash to\; H\; \backslash \backslash \; x\backslash mapsto\; \backslash prod\_\{i=1\}^n\; u\_x(i)\backslash end\{cases\}\; \backslash qquad\; \backslash begin\{cases\}\; P:\; G\backslash to\; H\; \backslash \backslash \; x\backslash mapsto\backslash prod\_\{i=1\}^n\; w\_x(i)\; \backslash end\{cases\}$

the pre-transfer from $G$ to $H$. The pre-transfer can be composed with a homomorphism $\backslash phi:\; H\backslash to\; A$ from $H$ into an abelian group $A$ to define a more general version of the transfer from $G$ to $A$ via $\backslash phi$, which occurs in the book by Gorenstein.

:$\backslash begin\{cases\}\; (\backslash phi\backslash circ\; P):\; G\backslash to\; A\; \backslash \backslash \; x\backslash mapsto\backslash prod\_\{i=1\}^n\; \backslash phi(u\_x(i))\backslash end\{cases\}\; \backslash qquad\; \backslash begin\{cases\}\; (\backslash phi\backslash circ\; P):\; G\backslash to\; A\; \backslash \backslash \; x\backslash mapsto\backslash prod\_\{i=1\}^n\; \backslash phi(w\_x(i))\backslash end\{cases\}$

Taking the natural epimorphism

:$\backslash begin\{cases\}\; \backslash phi:\; H\backslash to\; H/H\text{'}\; \backslash \backslash \; v\backslash mapsto\; vH\text{'}\; \backslash end\{cases\}$

yields the preceding definition of the Artin transfer $T\_\{G,H\}$ in its original form by Schur and by Emil Artin, which has also been dubbed Verlagerung by Hasse.{{cite journal|author=Hasse, H.|year=1930| title=Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz|journal=Jahresber. Deutsch. Math. Verein., Ergänzungsband|volume=6|pages=1–204}} Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism.

Proposition.{{cite book|author=Gorenstein, D.| year=2012| title=Finite groups| publisher=AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island}}{{cite book |author=Hall M., jr.|year=1999|title=The theory of groups|publisher=AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island}} The Artin transfers with respect to any two left transversals of $H$ in $G$ coincide.

:Proof. Let $(\backslash ell)\; =\; (\backslash ell\_1,\backslash ldots,\backslash ell\_n)$ and $(g)=(g\_1,\backslash ldots,g\_n)$ be two left transversals of $H$ in $G$. Then there exists a unique permutation $\backslash sigma\backslash in\; S\_n$ such that:

::$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; g\_iH=\backslash ell\_\{\backslash sigma(i)\}H.$

:Consequently:

::$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \},\; \backslash exists\; h\_i\; \backslash in\; H:\; \backslash qquad\; g\_ih\_i=\backslash ell\_\{\backslash sigma(i)\}.$

:For a fixed element $x\backslash in\; G$, there exists a unique permutation $\backslash lambda\_x\; \backslash in\; S\_n$ such that:

::$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; \backslash ell\_\{\backslash lambda\_x(\backslash sigma(i))\}H=x\backslash ell\_\{\backslash sigma(i)\}H=\; xg\_ih\_iH=\; xg\_iH=\; g\_\{\backslash pi\_x(i)\}H=\; g\_\{\backslash pi\_x(i)\}\; h\_\{\backslash pi\_x(i)\}H\; =\backslash ell\_\{\backslash sigma(\backslash pi\_x(i))\}\; H.$

:Therefore, the permutation representation of $G$ with respect to $(\backslash ell\_1,\; \backslash ldots,\; \backslash ell\_n)$ is given by $\backslash lambda\_x\; \backslash circ\; \backslash sigma\; =\; \backslash sigma\backslash circ\backslash pi\_x$ which yields: $\backslash lambda\_x\; =\backslash sigma\; \backslash circ\; \backslash pi\_x\; \backslash circ\; \backslash sigma^\{-1\}\; \backslash in\; S\_n.$ Furthermore, for the connection between the two elements:

::$\backslash begin\{align\}$

v_x(i) &:= \ell_{\lambda_x(i)}^{-1} x\ell_i\in H \\

u_x(i) &:= g_{\pi_x(i)}^{-1}xg_i\in H

\end{align}

:we have:

::$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; v\_x(\backslash sigma(i))\; =\backslash ell\_\{\backslash lambda\_x(\backslash sigma(i))\}^\{-1\}\; x\backslash ell\_\{\backslash sigma(i)\}=\; \backslash ell\_\{\backslash sigma(\backslash pi\_x(i))\}^\{-1\}\; xg\_ih\_i=\backslash left\; (g\_\{\backslash pi\_x(i)\}h\_\{\backslash pi\_x(i)\}\; \backslash right\; )^\{-1\}\; xg\_ih\_i=\; h\_\{\backslash pi\_x(i)\}^\{-1\}\; g\_\{\backslash pi\_x(i)\}^\{-1\}\; xg\_ih\_i=h\_\{\backslash pi\_x(i)\}^\{-1\}u\_x(i)h\_i.$

:Finally since $H/H\text{'}$ is abelian and $\backslash sigma$ and $\backslash pi\_x$ are permutations, the Artin transfer turns out to be independent of the left transversal:

::$T\_\{G,H\}^\{(\backslash ell)\}(x)=\backslash prod\_\{i=1\}^n\; v\_x(\backslash sigma(i))\backslash cdot\; H\text{'}=\backslash prod\_\{i=1\}^n\; h\_\{\backslash pi\_x(i)\}^\{-1\}u\_x(i)h\_i\backslash cdot\; H\text{'}=\backslash prod\_\{i=1\}^n\; u\_x(i)\backslash prod\_\{i=1\}^n\; h\_\{\backslash pi\_x(i)\}^\{-1\}\; \backslash prod\_\{i=1\}^n\; h\_i\backslash cdot\; H\text{'}=\backslash prod\_\{i=1\}^n\; u\_x(i)\backslash cdot\; 1\backslash cdot\; H\text{'}=\backslash prod\_\{i=1\}^n\; u\_x(i)\backslash cdot\; H\text{'}=T\_\{G,H\}^\{(g)\}(x),$

:as defined in formula (5).

Proposition. The Artin transfers with respect to any two right transversals of $H$ in $G$ coincide.

:Proof. Similar to the previous proposition.

Proposition. The Artin transfers with respect to $(g^\{-1\})\; =\; (g\_1^\{-1\},\backslash ldots,g\_n^\{-1\})$ and $(g)\; =\; (g\_1,\backslash ldots,g\_n)$ coincide.

:Proof. Using formula (4) and $H/H\text{'}$ being abelian we have:

::$T\_\{G,H\}^\{(g^\{-1\})\}(x)=\backslash prod\_\{i=1\}^n\; g\_i^\{-1\}xg\_\{\backslash rho\_x(i)\}\backslash cdot\; H\text{'}=\backslash prod\_\{i=1\}^n\; w\_x(i)\backslash cdot\; H\text{'}\; =\backslash prod\_\{i=1\}^n\; u\_\{x^\{-1\}\}(i)^\{-1\}\backslash cdot\; H\text{'}=\; \backslash left\; (\backslash prod\_\{i=1\}^n\; u\_\{x^\{-1\}\}(i)\backslash cdot\; H\text{'}\; \backslash right\; )^\{-1\}\; =\; \backslash left\; (T\_\{G,H\}^\{(g)\}\; \backslash left\; (x^\{-1\}\; \backslash right\; )\; \backslash right\; )^\{-1\}=T\_\{G,H\}^\{(g)\}(x).$

:The last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following section.

Corollary. The Artin transfer is independent of the choice of transversals and only depends on $H$ and $G$.

Theorem.{{cite book|author=Isaacs, I. M.|year=2008|title=Finite group theory| publisher= Graduate Studies in Mathematics, Vol. 92, American Mathematical Society, Providence, Rhode Island}}{{cite book| author=Aschbacher, M.|year=1986|title=Finite group theory|publisher= Cambridge Studies in Advanced Mathematics, Vol. 10, Cambridge University Press}}{{cite book|author=Smith, G. |author2=Tabachnikova, O.|year=2000| title=Topics in group theory|publisher=Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag, London}} Let $(g\_1,\backslash ldots,g\_n)$ be a left transversal of $H$ in $G$. The Artin transfer

:$\backslash begin\{cases\}\; T\_\{G,H\}:\; G\backslash to\; H/H\text{'}\; \backslash \backslash \; x\backslash mapsto\backslash prod\_\{i=1\}^n\; g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i\backslash cdot\; H\text{'}\; \backslash end\{cases\}$

and the permutation representation:

:$\backslash begin\{cases\}G\backslash to\; S\_n\; \backslash \backslash \; x\backslash mapsto\backslash pi\_x\; \backslash end\{cases\}$

are group homomorphisms:

:$(7)\backslash quad\; \backslash forall\; x,\; y\; \backslash in\; G:\; \backslash qquad\; T\_\{G,H\}(xy)=T\_\{G,H\}(x)\backslash cdot\; T\_\{G,H\}(y)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash pi\_\{xy\}=\backslash pi\_x\backslash circ\backslash pi\_y.$

{{hidden begin

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|titlestyle = background:LightSteelBlue; text-align:center;

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Let $x,y\backslash in\; G$:

:$T\_\{G,H\}(x)\backslash cdot\; T\_\{G,H\}(y)\; =\; \backslash prod\_\{i=1\}^n\; g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_iH\text{'}\backslash cdot\backslash prod\_\{j=1\}^n\; g\_\{\backslash pi\_y(j)\}^\{-1\}yg\_j\backslash cdot\; H\text{'}$

Since $H/H\text{'}$ is abelian and $\backslash pi\_y$ is a permutation, we can change the order of the factors in the product:

:$\backslash begin\{align\}$

\prod_{i=1}^n g_{\pi_x(i)}^{-1}xg_iH'\cdot\prod_{j=1}^n g_{\pi_y(j)}^{-1}yg_j\cdot H' &=\prod_{j=1}^n g_{\pi_x(\pi_y(j))}^{-1} x g_{\pi_y(j)} H'\cdot\prod_{j=1}^n g_{\pi_y(j)}^{-1}yg_j\cdot H' \\

&=\prod_{j=1}^n g_{\pi_x(\pi_y(j))}^{-1}xg_{\pi_y(j)}g_{\pi_y(j)}^{-1}yg_j\cdot H' \\

&=\prod_{j=1}^n g_{(\pi_x\circ\pi_y)(j))}^{-1}xyg_j\cdot H' \\

&=T_{G,H}(xy)

\end{align}

This relation simultaneously shows that the Artin transfer and the permutation representation are homomorphisms.

{{hidden end}}

It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial representation. The images of the factors $x,y$ are given by

:$T\_\{G,H\}(x)=\backslash prod\_\{i=1\}^n\; u\_x(i)\backslash cdot\; H\text{'}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; T\_\{G,H\}(y)=\backslash prod\_\{j=1\}^n\; u\_y(j)\backslash cdot\; H\text{'}.$

In the last proof, the image of the product $xy$ turned out to be

:$T\_\{G,H\}(xy)=\backslash prod\_\{j=1\}^n\; g\_\{\backslash pi\_x(\backslash pi\_y(j))\}^\{-1\}xg\_\{\backslash pi\_y(j)\}g\_\{\backslash pi\_y(j)\}^\{-1\}yg\_j\backslash cdot\; H\text{'}=\backslash prod\_\{j=1\}^n\; u\_x(\backslash pi\_y(j))\backslash cdot\; u\_y(j)\backslash cdot\; H\text{'}$,

which is a very peculiar law of composition discussed in more detail in the following section.

The law is reminiscent of crossed homomorphisms $x\backslash mapsto\; u\_x$ in the first cohomology group $\backslash mathrm\{H\}^1(G,M)$ of a $G$-module $M$, which have the property $u\_\{xy\}=u\_x^y\backslash cdot\; u\_y$ for $x,y\backslash in\; G$.

The peculiar structures which arose in the previous section can also be interpreted by endowing the cartesian product $H^n\backslash times\; S\_n$ with a special law of composition known as the wreath product $H\backslash wr\; S\_n$ of the groups $H$ and $S\_n$ with respect to the set $\backslash \{1,\; \backslash ldots,\; n\backslash \}.$

Definition. For $x,y\backslash in\; G$, the wreath product of the associated monomials and permutations is given by

:$(8)\; \backslash quad\; (u\_x(1),\backslash ldots,u\_x(n);\backslash pi\_x)\; \backslash cdot\; (u\_y(1),\backslash ldots,u\_y(n);\backslash pi\_y):=(u\_x(\backslash pi\_y(1))\backslash cdot\; u\_y(1),\backslash ldots,u\_x(\backslash pi\_y(n))\backslash cdot\; u\_y(n);\; \backslash pi\_x\; \backslash circ\backslash pi\_y)=(u\_\{xy\}(1),\backslash ldots,u\_\{xy\}(n);\backslash pi\_\{xy\}).$

Theorem. With this law of composition on $H^n\backslash times\; S\_n$ the monomial representation

:$\backslash begin\{cases\}\; G\backslash to\; H\backslash wr\; S\_n\; \backslash \backslash \; x\backslash mapsto\; (u\_x(1),\backslash ldots,u\_x(n);\backslash pi\_x)\; \backslash end\{cases\}$

is an injective homomorphism.

{{hidden begin

|title = Proof

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}}

The homomorphism property has been shown above already. For a homomorphism to be injective, it suffices to show the triviality of its kernel. The neutral element of the group $H^n\backslash times\; S\_n$ endowed with the wreath product is given by $(1,\backslash ldots,1;1)$, where the last $1$ means the identity permutation. If $(u\_x(1),\backslash ldots,u\_x(n);\backslash pi\_x)=(1,\backslash ldots,1;1)$, for some $x\backslash in\; G$, then $\backslash pi\_x=1$ and consequently

:$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}:\; \backslash qquad\; 1=u\_x(i)=g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i=g\_i^\{-1\}xg\_i.$

Finally, an application of the inverse inner automorphism with $g\_i$ yields $x\; =1$, as required for injectivity.

{{hidden end}}

Remark. The monomial representation of the theorem stands in contrast to the permutation representation, which cannot be injective if $|G|\; >\; n!.$

Remark. Whereas Huppert uses the monomial representation for defining the Artin transfer, we prefer to give the immediate definitions in formulas (5) and (6) and to merely illustrate the homomorphism property of the Artin transfer with the aid of the monomial representation.

Theorem. Let $G$ be a group with nested subgroups $K\backslash le\; H\backslash le\; G$ such that $(G:H)=n,\; (H:K)=m$ and Then the Artin transfer $T\_\{G,K\}$ is the compositum of the induced transfer $\backslash tilde\{T\}\_\{H,K\}:\; H/H\text{'}\backslash to\; K/K\text{'}$ and the Artin transfer $T\_\{G,H\}$, that is:

:$(9)\backslash quad\; T\_\{G,K\}=\backslash tilde\{T\}\_\{H,K\}\backslash circ\; T\_\{G,H\}$.

{{hidden begin

|title = Proof

|titlestyle = background:LightSteelBlue; text-align:center;

}}

If $(g\_1,\backslash ldots,g\_n)$ is a left transversal of $H$ in $G$ and $(h\_1,\backslash ldots,h\_m)$ is a left transversal of $K$ in $H$, that is $G=\backslash sqcup\_\{i=1\}^n\; g\_iH$ and $H=\backslash sqcup\_\{j=1\}^m\; h\_jK$, then

:$G=\backslash bigsqcup\_\{i=1\}^n\; \backslash bigsqcup\_\{j=1\}^m\; g\_ih\_jK$

is a disjoint left coset decomposition of $G$ with respect to $K$.

Given two elements $x\backslash in\; G$ and $y\backslash in\; H$, there exist unique permutations $\backslash pi\_x\backslash in\; S\_n$, and $\backslash sigma\_y\backslash in\; S\_m$, such that

:$\backslash begin\{align\}$

u_x(i) & :=g_{\pi_x(i)}^{-1}xg_i\in H && \text{for all } 1\le i\le n \\

v_y(j) & :=h_{\sigma_y(j)}^{-1}yh_j\in K && \text{for all } 1\le j\le m

\end{align}

Then, anticipating the definition of the induced transfer, we have

:$\backslash begin\{align\}$

T_{G,H}(x) &=\prod_{i=1}^n u_x(i)\cdot H' \\

\tilde{T}_{H,K}(y\cdot H') &=T_{H,K}(y)=\prod_{j=1}^m v_y(j)\cdot K'

\end{align}

For each pair of subscripts $1\backslash le\; i\backslash le\; n$ and $1\backslash le\; j\backslash le\; m$, we put $y\_i:=u\_x(i)$, and we obtain

:$xg\_ih\_j=g\_\{\backslash pi\_x(i)\}g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_ih\_j=g\_\{\backslash pi\_x(i)\}u\_x(i)h\_j=g\_\{\backslash pi\_x(i)\}y\_ih\_j\; =g\_\{\backslash pi\_x(i)\}h\_\{\backslash sigma\_\{y\_i\}(j)\}h\_\{\backslash sigma\_\{y\_i\}(j)\}^\{-1\}\; y\_ih\_j\; =g\_\{\backslash pi\_x(i)\}h\_\{\backslash sigma\_\{y\_i\}(j)\}v\_\{y\_i\}(j),$

resp.

:$h\_\{\backslash sigma\_\{y\_i\}(j)\}^\{-1\}g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_ih\_j=v\_\{y\_i\}(j).$

Therefore, the image of $x$ under the Artin transfer $T\_\{G,K\}$ is given by

:$\backslash begin\{align\}$

T_{G,K}(x) &=\prod_{i=1}^n \prod_{j=1}^m v_{y_i}(j)\cdot K' \\

&=\prod_{i=1}^n \prod_{j=1}^m h_{\sigma_{y_i}(j)}^{-1}g_{\pi_x(i)}^{-1}xg_ih_j\cdot K' \\

&=\prod_{i=1}^n \prod_{j=1}^m h_{\sigma_{y_i}(j)}^{-1}u_x(i)h_j\cdot K' \\

&=\prod_{i=1}^n \prod_{j=1}^m h_{\sigma_{y_i}(j)}^{-1}y_ih_j\cdot K' \\

&=\prod_{i=1}^n \tilde{T}_{H,K} \left (y_i\cdot H' \right ) \\

&=\tilde{T}_{H,K} \left (\prod_{i=1}^n y_i\cdot H' \right ) \\

&=\tilde{T}_{H,K} \left (\prod_{i=1}^n u_x(i)\cdot H' \right ) \\

&=\tilde{T}_{H,K}(T_{G,H}(x))

\end{align}

{{hidden end}}

Finally, we want to emphasize the structural peculiarity of the monomial representation

:$\backslash begin\{cases\}G\backslash to\; K^\{n\backslash cdot\; m\}\backslash times\; S\_\{n\backslash cdot\; m\}\; \backslash \backslash \; x\backslash mapsto\; (k\_x(1,1),\backslash ldots,k\_x(n,m);\backslash gamma\_x)\; \backslash end\{cases\}$

which corresponds to the composite of Artin transfers, defining

:$k\_x(i,j):=\; \backslash left\; ((gh)\_\{\backslash gamma\_x(i,j)\}\; \backslash right)^\{-1\}x(gh)\_\{(i,j)\}\backslash in\; K$

for a permutation $\backslash gamma\_x\backslash in\; S\_\{n\backslash cdot\; m\}$, and using the symbolic notation $(gh)\_\{(i,j)\}:=g\_ih\_j$ for all pairs of subscripts $1\backslash le\; i\backslash le\; n$, $1\backslash le\; j\backslash le\; m$.

The preceding proof has shown that

:$k\_x(i,j)=h\_\{\backslash sigma\_\{y\_i\}(j)\}^\{-1\}g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_ih\_j.$

Therefore, the action of the permutation $\backslash gamma\_x$ on the set $[1,n]\backslash times[1,m]$ is given by $\backslash gamma\_x\; (i,j)\; =(\backslash pi\_x(i),\; \backslash sigma\_\{u\_x(i)\}(j))$. The action on the second component $j$ depends on the first component $i$ (via the permutation $\backslash sigma\_\{u\_x(i)\}\backslash in\; S\_m$), whereas the action on the first component $i$ is independent of the second component $j$. Therefore, the permutation $\backslash gamma\_x\backslash in\; S\_\{n\backslash cdot\; m\}$ can be identified with the multiplet

:$(\backslash pi\_x;\backslash sigma\_\{u\_x(1)\},\backslash ldots,\backslash sigma\_\{u\_x(n)\})\backslash in\; S\_n\backslash times\; S\_m^n,$

which will be written in twisted form in the next section.

The permutations $\backslash gamma\_x$, which arose as second components of the monomial representation

:$\backslash begin\{cases\}\; G\backslash to\; K\backslash wr\; S\_\{n\backslash cdot\; m\}\; \backslash \backslash \; x\backslash mapsto\; (k\_x(1,1),\backslash ldots,k\_x(n,m);\backslash gamma\_x)\; \backslash end\{cases\}$

in the previous section, are of a very special kind. They belong to the stabilizer of the natural equipartition of the set $[1,n]\backslash times[1,m]$ into the $n$ rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product $S\_m\backslash wr\; S\_n$ of the symmetric groups $S\_m$ and $S\_n$ with respect to the set $\backslash \{\; 1,\backslash ldots,n\backslash \}$, whose underlying set $S\_m^n\backslash times\; S\_n$ is endowed with the following law of composition:

:$\backslash begin\{align\}$

(10)\quad \forall x,z\in G: \qquad \gamma_x\cdot\gamma_z &=(\sigma_{u_x(1)},\ldots,\sigma_{u_x(n)};\pi_x)\cdot (\sigma_{u_z(1)},\ldots,\sigma_{u_z(n)};\pi_z)\\

&=(\sigma_{u_x(\pi_z(1))}\circ\sigma_{u_z(1)},\ldots,\sigma_{u_x(\pi_z(n))}\circ\sigma_{u_z(n)};\pi_x\circ \pi_z) \\

&=(\sigma_{u_{xz}(1)},\ldots,\sigma_{u_{xz}(n)};\pi_{xz}) \\

&=\gamma_{xz}

\end{align}

This law reminds of the chain rule $D(g\backslash circ\; f)(x)=D(g)(f(x))\backslash circ\; D(f)(x)$ for the Fréchet derivative in $x\backslash in\; E$ of the compositum of differentiable functions $f:\; E\backslash to\; F$ and $g:\; F\backslash to\; G$ between complete normed spaces.

The above considerations establish a third representation, the stabilizer representation,

:$\backslash begin\{cases\}\; G\backslash to\; S\_m\backslash wr\; S\_n\; \backslash \backslash \; x\backslash mapsto(\backslash sigma\_\{u\_x(1)\},\backslash ldots,\backslash sigma\_\{u\_x(n)\};\backslash pi\_x)\; \backslash end\{cases\}$

of the group $G$ in the wreath product $S\_m\backslash wr\; S\_n$, similar to the permutation representation and the monomial representation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, certainly not, if $G$ is infinite. Formula (10) proves the following statement.

Theorem. The stabilizer representation

:$\backslash begin\{cases\}\; G\backslash to\; S\_m\backslash wr\; S\_n\; \backslash \backslash \; x\backslash mapsto\backslash gamma\_\{x\}=(\backslash sigma\_\{u\_x(1)\},\backslash ldots,\backslash sigma\_\{u\_x(n)\};\backslash pi\_x)\; \backslash end\{cases\}$

of the group $G$ in the wreath product $S\_m\backslash wr\; S\_n$ of symmetric groups is a group homomorphism.

Let $(g\_1,\backslash ldots,g\_n)$ be a left transversal of a subgroup $H$ of finite index $n$ in a group $G$ and $x\; \backslash mapsto\; \backslash pi\_x$ be its associated permutation representation.

Theorem. Suppose the permutation $\backslash pi\_x$ decomposes into pairwise disjoint (and thus commuting) cycles $\backslash zeta\_1,\; \backslash ldots,\; \backslash zeta\_t\; \backslash in\; S\_n$ of lengths $f\_1,\; \backslash ldots\; f\_t,$ which is unique up to the ordering of the cycles. More explicitly, suppose

:$(11)\backslash quad\; \backslash left\; (g\_jH,\; g\_\{\backslash zeta\_j(j)\}H,\; g\_\{\backslash zeta\_j^2(j)\}H,\; \backslash ldots,\; g\_\{\backslash zeta\_j^\{f\_j-1\}(j)\}H\; \backslash right\; )=\; \backslash left\; (g\_jH,\; xg\_jH,\; x^2g\_jH,\; \backslash ldots,\; x^\{f\_j-1\}\; g\_jH\; \backslash right\; ),$

for $1\backslash le\; j\backslash le\; t$, and $f\_1\; +\; \backslash cdots+\; f\_t=n.$ Then the image of $x\backslash in\; G$ under the Artin transfer is given by

:$(12)\backslash quad\; T\_\{G,H\}(x)=\backslash prod\_\{j=1\}^t\; g\_j^\{-1\}x^\{f\_j\}g\_j\backslash cdot\; H\text{'}.$

{{hidden begin

|title = Proof

|titlestyle = background:LightSteelBlue; text-align:center;

}}

Define $\backslash ell\_\{j,k\}\; :=x^kg\_j$ for $0\backslash le\; k\backslash le\; f\_j-1$ and $1\backslash le\; j\backslash le\; t$. This is a left transversal of $H$ in $G$ since

:$(13)\backslash quad\; G=\backslash bigsqcup\_\{j=1\}^t\; \backslash bigsqcup\_\{k=0\}^\{f\_j-1\}\; x^kg\_jH$

is a disjoint decomposition of $G$ into left cosets of $H$.

Fix a value of $1\backslash le\; j\backslash le\; t$. Then:

:$\backslash begin\{align\}$

x\ell_{j,k} &=xx^kg_j=x^{k+1}g_j=\ell_{j,k+1}\in\ell_{j,k+1}H && \forall k \in \{0, \ldots, f_j -2\} \\

x\ell_{j,f_j-1} &=xx^{f_j-1}g_j=x^{f_j}g_j\in g_jH=\ell_{j,0}H

\end{align}

Define:

:$\backslash begin\{align\}$

u_x(j,k) &:=\ell_{j,k+1}^{-1}x\ell_{j,k}=1\in H && \forall k \in \{0, \ldots, f_j -2\} \\

u_x(j,f_j-1) &:=\ell_{j,0}^{-1}x\ell_{j,f_j-1}=g_j^{-1}x^{f_j}g_j\in H

\end{align}

Consequently,

:$T\_\{G,H\}(x)=\backslash prod\_\{j=1\}^t\; \backslash prod\_\{k=0\}^\{f\_j-1\}\; u\_x(j,k)\backslash cdot\; H\text{'}=\backslash prod\_\{j=1\}^t\; \backslash left\; (\backslash prod\_\{k=0\}^\{f\_j-2\}\; 1\; \backslash right\; )\backslash cdot\; u\_x(j,f\_j-1)\backslash cdot\; H\text{'}=\backslash prod\_\{j=1\}^t\; g\_j^\{-1\}x^\{f\_j\}g\_j\backslash cdot\; H\text{'}.$

{{hidden end}}

The cycle decomposition corresponds to a $(\backslash langle\; x\backslash rangle,\; H)$ double coset decomposition of $G$:

:$G=\backslash bigsqcup\_\{j=1\}^t\; \backslash langle\; x\backslash rangle\; g\_jH$

It was this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper.

Let $H$ be a normal subgroup of finite index $n$ in a group $G$. Then we have $xH=\; Hx$, for all $x\backslash in\; G$, and there exists the quotient group $G/H$ of order $n$. For an element $x\backslash in\; G$, we let $f:=\; \backslash mathrm\{ord\}\; (xH)$ denote the order of the coset $xH$ in $G/H$, and we let $(g\_1,\backslash ldots,g\_t)$ be a left transversal of the subgroup $\backslash langle\; x,H\backslash rangle$ in $G$, where $t=n/f$.

Theorem. Then the image of $x\backslash in\; G$ under the Artin transfer $T\_\{G,H\}$ is given by:

:$(14)\backslash quad\; T\_\{G,H\}(x)=\backslash prod\_\{j=1\}^t\; g\_j^\{-1\}x^fg\_j\backslash cdot\; H\text{'}$.

{{hidden begin

|title = Proof

|titlestyle = background:LightSteelBlue; text-align:center;

}}

$\backslash langle\; xH\backslash rangle$ is a cyclic subgroup of order $f$ in $G/H$, and a left transversal $(g\_1,\backslash ldots,g\_t)$ of the subgroup $\backslash langle\; x,H\backslash rangle$ in $G$, where $t=n/f$ and $G=\backslash sqcup\_\{j=1\}^t\; g\_j\backslash langle\; x,H\backslash rangle$ is the corresponding disjoint left coset decomposition, can be refined to a left transversal $g\_jx^k\; (1\backslash le\; j\backslash le\; t,\backslash \; 0\backslash le\; k\backslash le\; f-1)$ with disjoint left coset decomposition:

:$(15)\backslash quad\; G=\backslash sqcup\_\{j=1\}^t\; \backslash sqcup\_\{k=0\}^\{f-1\}\; g\_jx^kH$

of $H$ in $G$. Hence, the formula for the image of $x$ under the Artin transfer $T\_\{G,H\}$ in the previous section takes the particular shape

:$T\_\{G,H\}(x)=\backslash prod\_\{j=1\}^t\; g\_j^\{-1\}x^fg\_j\backslash cdot\; H\text{'}$

with exponent $f$ independent of $j$.

{{hidden end}}

Corollary. In particular, the inner transfer of an element $x\backslash in\; H$ is given as a symbolic power:

:$(16)\backslash quad\; T\_\{G,H\}(x)=x^\{\backslash mathrm\{Tr\}\_G(H)\}\backslash cdot\; H\text{'}$

with the trace element

:$(17)\backslash quad\; \backslash mathrm\{Tr\}\_G(H)=\backslash sum\_\{j=1\}^t\; g\_j\backslash in\backslash Z[G]$

of $H$ in $G$ as symbolic exponent.

The other extreme is the outer transfer of an element $x\backslash in\; G\backslash setminus\; H$ which generates $G/H$, that is $G=\backslash langle\; x,H\backslash rangle$.

It is simply an $n$th power

:$(18)\backslash quad\; T\_\{G,H\}(x)=x^n\backslash cdot\; H\text{'}$.

{{hidden begin

|title = Proof

|titlestyle = background:LightSteelBlue; text-align:center;

}}

The inner transfer of an element $x\backslash in\; H$, whose coset $xH=H$ is the principal set in $G/H$ of order $f=1$, is given as the symbolic power

:$T\_\{G,H\}(x)=\backslash prod\_\{j=1\}^t\; g\_j^\{-1\}xg\_j\backslash cdot\; H\text{'}=\backslash prod\_\{j=1\}^t\; x^\{g\_j\}\backslash cdot\; H\text{'}=x^\{\backslash sum\_\{j=1\}^t\; g\_j\}\backslash cdot\; H\text{'}$

with the trace element

:$\backslash mathrm\{Tr\}\_G(H)=\backslash sum\_\{j=1\}^t\; g\_j\backslash in\backslash Z[G]$

of $H$ in $G$ as symbolic exponent.

The outer transfer of an element $x\backslash in\; G\backslash setminus\; H$ which generates $G/H$, that is $G=\backslash langle\; x,H\; \backslash rangle$, whence the coset $xH$ is generator of $G/H$ with order$f=n$, is given as the $n$th power

:$T\_\{G,H\}(x)=\backslash prod\_\{j=1\}^1\; 1^\{-1\}\backslash cdot\; x^n\backslash cdot\; 1\backslash cdot\; H\text{'}=x^n\backslash cdot\; H\text{'}.$

{{hidden end}}

Transfers to normal subgroups will be the most important cases in the sequel, since the central concept of this article, the Artin pattern, which endows descendant trees with additional structure, consists of targets and kernels of Artin transfers from a group $G$ to intermediate groups $G\text{'}\backslash le\; H\backslash le\; G$ between $G$ and $G\text{'}$. For these intermediate groups we have the following lemma.

Lemma. All subgroups containing the commutator subgroup are normal.

{{hidden begin

|title = Proof

|titlestyle = background:LightSteelBlue; text-align:center;

}}

Let $G\text{'}\backslash le\; H\backslash le\; G$. If $H$ were not a normal subgroup of $G$, then we had $x^\{-1\}Hx\backslash not\backslash subseteq\; H$ for some element $x\backslash in\; G\backslash setminus\; H$. This would imply the existence of elements $h\backslash in\; H$ and $y\backslash in\; G\backslash setminus\; H$ such that $x^\{-1\}hx=y$, and consequently the commutator $[h,x]=\; h^\{-1\}x^\{-1\}hx\; =h^\{-1\}y$ would be an element in $G\backslash setminus\; H$ in contradiction to $G\text{'}\backslash le\; H$.

{{hidden end}}

Explicit implementations of Artin transfers in the simplest situations are presented in the following section.

Let $G$ be a p-group with abelianization $G/G\text{'}$ of elementary abelian type $(p,p)$. Then $G$ has $p+1$ maximal subgroups $H\_1,\; \backslash ldots,\; H\_\{p+1\}$ of index $p.$

Lemma. In this particular case, the Frattini subgroup, which is defined as the intersection of all maximal subgroups coincides with the commutator subgroup.

Proof. To see this note that due to the abelian type of $G/G\text{'}$ the commutator subgroup contains all p-th powers $G\text{'}\backslash supset\; G^p,$ and thus we have $\backslash Phi(G)=G^p\backslash cdot\; G\text{'}=G\text{'}$.

For each $1\backslash le\; i\backslash le\; p+1$, let $T\_i:\; G\backslash to\; H\_i/H\_i\text{'}$ be the Artin transfer homomorphism. According to Burnside's basis theorem the group $G$ can therefore be generated by two elements $x,y$ such that $x^p,y^p\backslash in\; G\text{'}.$ For each of the maximal subgroups $H\_i$, which are also normal we need a generator $h\_i$ with respect to $G\text{'}$, and a generator $t\_i$ of a transversal $(1,t\_i,t\_i^2,\backslash ldots,t\_i^\{p-1\})$ such that

:$\backslash begin\{align\}$

H_i &= \langle h_i,G' \rangle \\

G &= \langle t_i,H_i\rangle=\bigsqcup_{j=0}^{p-1} t_i^jH_i

\end{align}

A convenient selection is given by

:

Then, for each $1\backslash le\; i\backslash le\; p+1$ we use equations (16) and (18) to implement the inner and outer transfers:

:$\backslash begin\{align\}$

(20)\quad T_i(h_i) &= h_i^{\mathrm{Tr}_G(H_i)}\cdot H_i'=h_i^{1+t_i+t_i^2+\cdots +t_i^{p-1}}\cdot H_i' = h_i\cdot \left ( t_i^{-1}h_it_i \right )\cdot \left ( t_i^{-2}h_it_i^2\right ) \cdots \left ( t_i^{-p+1}h_it_i^{p-1}\right ) \cdot H_i'= \left (h_it_i^{-1} \right )^p t_i^p\cdot H_i' \\

(21)\quad T_i(t_i) &= t_i^p\cdot H_i'

\end{align},

The reason is that in $G/H\_i,$ $\backslash mathrm\{ord\}(h\_iH\_i)=1$ and $\backslash mathrm\{ord\}(t\_iH\_i)=p.$

The complete specification of the Artin transfers $T\_i$ also requires explicit knowledge of the derived subgroups $H\_i\text{'}$. Since $G\text{'}$ is a normal subgroup of index $p$ in $H\_i$, a certain general reduction is possible by $H\_i\text{'}=[H\_i,H\_i]=[G\text{'},H\_i]=(G\text{'})^\{h\_i-1\},${{cite journal| author=Blackburn, N.| year=1958| title=On a special class of p-groups| journal=Acta Math.| volume=100| issue=1–2| pages=45–92 |doi= 10.1007/bf02559602| doi-access=free}} but a presentation of $G$ must be known for determining generators of $G\text{'}=\backslash langle\; s\_1,\; \backslash ldots,\; s\_n\backslash rangle$, whence

:$(22)\backslash quad\; H\_i\text{'}=(G\text{'})^\{h\_i-1\}=\backslash langle[\; s\_1,h\_i],\backslash ldots,[s\_n,h\_i]\backslash rangle.$

Let $G$ be a p-group with abelianization $G/G\text{'}$ of non-elementary abelian type $(p^2,p)$. Then $G$ has $p+1$ maximal subgroups $H\_1,\; \backslash ldots,\; H\_\{p+1\}$ of index $p$ and $p+1$ subgroups $U\_1,\; \backslash ldots,\; U\_\{p+1\}$ of index $p^2.$ For each $i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; p+1\backslash \}$ let

:$\backslash begin\{align\}$

T_{1,i}: G &\to H_i/H_i' \\

T_{2,i}: G &\to U_i/U_i'

\end{align}

be the Artin transfer homomorphisms. Burnside's basis theorem asserts that the group $G$ can be generated by two elements $x,y$ such that $x^\{p^2\},y^p\backslash in\; G\text{'}.$

We begin by considering the first layer of subgroups. For each of the normal subgroups $H\_i$, we select a generator

:$(23)\backslash quad\; h\_i=xy^\{i-1\}$

such that $H\_i=\backslash langle\; h\_i,G\text{'}\backslash rangle$. These are the cases where the factor group $H\_i/G\text{'}$ is cyclic of order $p^2$. However, for the distinguished maximal subgroup $H\_\{p+1\}$, for which the factor group $H\_\{p+1\}/G\text{'}$ is bicyclic of type $(p,p)$, we need two generators:

:$(24)\backslash quad\; \backslash begin\{cases\}\; h\_\{p+1\}=y\; \backslash \backslash \; h\_0=x^p\; \backslash end\{cases\}$

such that $H\_\{p+1\}=\backslash langle\; h\_\{p+1\},h\_0,G\text{'}\backslash rangle$. Further, a generator $t\_i$ of a transversal must be given such that $G=\backslash langle\; t\_i,H\_i\backslash rangle$, for each $1\backslash le\; i\backslash le\; p+1$. It is convenient to define

:

Then, for each $1\backslash le\; i\backslash le\; p+1$, we have inner and outer transfers:

:$\backslash begin\{align\}$

(26)\quad T_{1,i}(h_i) &=h_i^{\mathrm{Tr}_G(H_i)}\cdot H_i'=h_i^{1+t_i+t_i^2+\ldots +t_i^{p-1}}\cdot H_i' = \left (h_it_i^{-1} \right )^pt_i^p\cdot H_i' \\

(27)\quad T_{1,i}(t_i) &=t_i^p\cdot H_i'

\end{align}

since $\backslash mathrm\{ord\}(h\_iH\_i)=1$ and $\backslash mathrm\{ord\}(t\_iH\_i)=p$.

Now we continue by considering the second layer of subgroups. For each of the normal subgroups $U\_i$, we select a generator

:

such that $U\_i=\backslash langle\; u\_i,G\text{'}\backslash rangle$. Among these subgroups, the Frattini subgroup $U\_\{p+1\}=\backslash langle\; x^p,\; G\text{'}\; \backslash rangle\; =G^p\backslash cdot\; G\text{'}$ is particularly distinguished. A uniform way of defining generators $t\_i,w\_i$ of a transversal such that $G=\backslash langle\; t\_i,w\_i,U\_i\backslash rangle$, is to set

:

Since $\backslash mathrm\{ord\}(u\_iU\_i)=1$, but on the other hand $\backslash mathrm\{ord\}(t\_iU\_i)=p^2$ and $\backslash mathrm\{ord\}(w\_iU\_i)=\; p$, for $1\backslash le\; i\backslash le\; p+1$, with the single exception that $\backslash mathrm\{ord\}(t\_\{p+1\}U\_\{p+1\})=p$, we obtain the following expressions for the inner and outer transfers

:$\backslash begin\{align\}$

(30)\quad T_{2,i}(u_i) &= u_i^{\mathrm{Tr}_G(U_i)}\cdot U_i'=u_i^{\sum_{j=0}^{p-1} \sum_{k=0}^{p-1} w_i^jt_i^k}\cdot U_i' =\prod_{j=0}^{p-1} \prod_{k=0}^{p-1} (w_i^jt_i^k)^{-1}u_iw_i^jt_i^k\cdot U_i' \\

(31)\quad T_{2,i}(t_i) &= t_i^{p^2}\cdot U_i'

\end{align}

exceptionally

:$\backslash begin\{align\}$

&(32)\quad T_{2,p+1} \left (t_{p+1} \right ) = \left (t_{p+1}^p \right )^{1+w_{p+1}+w_{p+1}^2+\ldots +w_{p+1}^{p-1}}\cdot U_{p+1}' \\

&(33)\quad T_{2,i}(w_i) = \left (w_i^p \right )^{1+t_i+t_i^2+\ldots +t_i^{p-1}}\cdot U_i' && 1\le i\le p+1

\end{align}

The structure of the derived subgroups $H\_i\text{'}$ and $U\_i\text{'}$ must be known to specify the action of the Artin transfers completely.

Let $G$ be a group with finite abelianization $G/G\text{'}$. Suppose that $(H\_i)\_\{i\backslash in\; I\}$ denotes the family of all subgroups which contain $G\text{'}$ and are therefore necessarily normal, enumerated by a finite index set $I$. For each $i\backslash in\; I$, let $T\_i:=T\_\{G,H\_i\}$ be the Artin transfer from $G$ to the abelianization $H\_i/H\_i\text{'}$.

Definition.{{cite journal| author=Mayer, D. C.| year=2013| title=The distribution of second p-class groups on coclass graphs|

journal=J. Théor. Nombres Bordeaux| volume=25| number=2| pages=401–456 |doi=10.5802/jtnb.842| arxiv=1403.3833| s2cid=62897311}} The family of normal subgroups $\backslash varkappa\_H(G)=(\backslash ker(T\_i))\_\{i\backslash in\; I\}$ is called the transfer kernel type (TKT) of $G$ with respect to $(H\_i)\_\{i\backslash in\; I\}$, and the family of abelianizations (resp. their abelian type invariants) $\backslash tau\_H(G)=(H\_i/H\_i\text{'})\_\{i\backslash in\; I\}$ is called the transfer target type (TTT) of $G$ with respect to $(H\_i)\_\{i\backslash in\; I\}$. Both families are also called multiplets whereas a single component will be referred to as a singulet.

Important examples for these concepts are provided in the following two sections.

Let $G$ be a p-group with abelianization $G/G\text{'}$ of elementary abelian type $(p,p)$. Then $G$ has $p+1$ maximal subgroups $H\_1,\; \backslash ldots,\; H\_\{p+1\}$ of index $p$. For $i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; p+1\backslash \}$ let $T\_i:\; G\backslash to\; H\_i/H\_i\text{'}$ denote the Artin transfer homomorphism.

Definition. The family of normal subgroups $\backslash varkappa\_H(G)=(\backslash ker(T\_i))\_\{1\backslash le\; i\backslash le\; p+1\}$ is called the transfer kernel type (TKT) of $G$ with respect to $H\_1,\backslash ldots,H\_\{p+1\}$.

Remark. For brevity, the TKT is identified with the multiplet $(\backslash varkappa(i))\_\{1\backslash le\; i\backslash le\; p+1\}$, whose integer components are given by

:

Here, we take into consideration that each transfer kernel $\backslash ker(T\_i)$ must contain the commutator subgroup $G\text{'}$ of $G$, since the transfer target $H\_i/H\_i\text{'}$ is abelian. However, the minimal case $\backslash ker(T\_i)=G\text{'}$ cannot occur.

Remark. A renumeration of the maximal subgroups $K\_i=H\_\{\backslash pi(i)\}$ and of the transfers $V\_i=T\_\{\backslash pi(i)\}$ by means of a permutation $\backslash pi\backslash in\; S\_\{p+1\}$ gives rise to a new TKT $\backslash lambda\_K(G)=(\backslash ker(V\_i))\_\{1\backslash le\; i\backslash le\; p+1\}$ with respect to $K\_1,\backslash ldots,K\_\{p+1\}$, identified with $(\backslash lambda(i))\_\{1\backslash le\; i\backslash le\; p+1\}$, where

:

It is adequate to view the TKTs $\backslash lambda\_K(G)\backslash sim\backslash varkappa\_H(G)$ as equivalent. Since we have

:$K\_\{\backslash lambda(i)\}\; =\; \backslash ker(V\_i)\; =\; \backslash ker(T\_\{\backslash pi(i)\})\; =\; H\_\{\backslash varkappa(\backslash pi(i))\}\; =\; K\_\{\backslash tilde\{\backslash pi\}^\{-1\}(\backslash varkappa(\backslash pi(i)))\},$

the relation between $\backslash lambda$ and $\backslash varkappa$ is given by $\backslash lambda\; =\; \backslash tilde\{\backslash pi\}^\{-1\}\; \backslash circ\; \backslash varkappa\; \backslash circ\; \backslash pi$. Therefore, $\backslash lambda$ is another representative of the orbit $\backslash varkappa^\{S\_\{p+1\}\}$ of $\backslash varkappa$ under the action $(\backslash pi,\backslash mu)\backslash mapsto\backslash tilde\{\backslash pi\}^\{-1\}\backslash circ\backslash mu\backslash circ\backslash pi$ of the symmetric group $S\_\{p+1\}$ on the set of all mappings from $\backslash \{\; 1,\backslash ldots,p+1\backslash \}\; \backslash to\; \backslash \{\; 0,1,\; \backslash ldots,p+1\backslash \},$ where the extension $\backslash tilde\{\backslash pi\}\backslash in\; S\_\{p+2\}$ of the permutation $\backslash pi\backslash in\; S\_\{p+1\}$ is defined by $\backslash tilde\{\backslash pi\}(0)=0,$ and formally $H\_0=G,\; K\_0=G.$

Definition. The orbit $\backslash varkappa(G)=\backslash varkappa^\{S\_\{p+1\}\}$ of any representative $\backslash varkappa$ is an invariant of the p-group $G$ and is called its transfer kernel type, briefly TKT.

Remark. Let $\backslash \#\backslash mathcal\{H\}\_0(G):=\backslash \#\backslash \{\; 1\backslash le\; i\backslash le\; p+1\backslash mid\backslash varkappa(i)=0\backslash \}$ denote the counter of total transfer kernels $\backslash ker(T\_i)=G$, which is an invariant of the group $G$. In 1980, S. M. Chang and R. Foote{{cite journal| author= Chang, S. M. |author2=Foote, R.| year=1980| title=Capitulation in class field extensions of type (p,p)| journal=Can. J. Math.| volume =32| number =5| pages=1229–1243 |doi=10.4153/cjm-1980-091-9| doi-access=free}} proved that, for any odd prime $p$ and for any integer $0\backslash le\; n\backslash le\; p+1$, there exist metabelian p-groups $G$ having abelianization $G/G\text{'}$ of type $(p,p)$ such that $\backslash \#\backslash mathcal\{H\}\_0(G)=n$. However, for $p=2$, there do not exist non-abelian $2$-groups $G$ with $G/G\text{'}\backslash simeq\; (2,2)$, which must be metabelian of maximal class, such that $\backslash \#\backslash mathcal\{H\}\_0(G)\backslash ge\; 2$. Only the elementary abelian $2$-group $G=C\_2\backslash times\; C\_2$ has $\backslash \#\backslash mathcal\{H\}\_0(G)=3$. See Figure 5.

In the following concrete examples for the counters $\backslash \#\backslash mathcal\{H\}\_0(G)$, and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by H. U. Besche, B. Eick and E. A. O'Brien.

{{cite book| author=Besche, H. U. |author2=Eick, B. |author3=O'Brien, E. A.| year=2005| title=The SmallGroups Library – a library of groups of small order| publisher=An accepted and refereed GAP 4 package, available also in MAGMA}}{{cite journal| author=Besche, H. U. |author2=Eick, B. |author3=O'Brien, E. A.| year=2002| title=A millennium project: constructing small groups| journal=Int. J. Algebra Comput.| volume=12|issue=5 |

pages=623–644 |doi=10.1142/s0218196702001115}}

For $p=3$, we have

• $\backslash \#\backslash mathcal\{H\}\_0(G)=0$ for the extra special group $G=\backslash langle\; 27,4\backslash rangle$ of exponent $9$ with TKT $\backslash varkappa=(1111)$ (Figure 6),
• $\backslash \#\backslash mathcal\{H\}\_0(G)=1$ for the two groups $G\backslash in\backslash \{\backslash langle\; 243,6\backslash rangle,\backslash langle\; 243,8\backslash rangle\backslash \}$ with TKTs $\backslash varkappa\backslash in\backslash \{\; (0122),(2034)\backslash \}$ (Figures 8 and 9),
• $\backslash \#\backslash mathcal\{H\}\_0(G)=2$ for the group $G=\backslash langle\; 243,3\backslash rangle$ with TKT $\backslash varkappa=(0043)$ (Figure 4 in the article on descendant trees),
• $\backslash \#\backslash mathcal\{H\}\_0(G)=3$ for the group $G=\backslash langle\; 81,7\backslash rangle$ with TKT $\backslash varkappa=(2000)$ (Figure 6),
• $\backslash \#\backslash mathcal\{H\}\_0(G)=4$ for the extra special group $G=\backslash langle\; 27,3\backslash rangle$ of exponent $3$ with TKT $\backslash varkappa=(0000)$ (Figure 6).

Let $G$ be a p-group with abelianization $G/G\text{'}$ of non-elementary abelian type $(p^2,p).$ Then $G$ possesses $p+1$ maximal subgroups $H\_1,\; \backslash ldots,\; H\_\{p+1\}$ of index $p$ and $p+1$ subgroups $U\_1,\; \backslash ldots,\; U\_\{p+1\}$ of index $p^2.$

Assumption. Suppose

:$H\_\{p+1\}=\backslash prod\_\{j=1\}^\{p+1\}\; U\_j$

is the distinguished maximal subgroup and

:$U\_\{p+1\}=\backslash bigcap\_\{j=1\}^\{p+1\}\; H\_j$

is the distinguished subgroup of index $p^2$ which as the intersection of all maximal subgroups, is the Frattini subgroup $\backslash Phi(G)$ of $G$.

For each $1\backslash le\; i\backslash le\; p+1$, let $T\_\{1,i\}:\; G\backslash to\; H\_i/H\_i\text{'}$ denote the Artin transfer homomorphism.

Definition. The family $\backslash varkappa\_\{1,H,U\}(G)=(\backslash ker(T\_\{1,i\}))\_\{i=1\}^\{p+1\}$ is called the first layer transfer kernel type of $G$ with respect to $H\_1,\backslash ldots,H\_\{p+1\}$ and $U\_1,\backslash ldots,U\_\{p+1\}$, and is identified with $(\backslash varkappa\_1(i))\_\{i=1\}^\{p+1\}$, where

:

Remark. Here, we observe that each first layer transfer kernel is of exponent $p$ with respect to $G\text{'}$ and consequently cannot coincide with $H\_j$ for any $1\backslash le\; j\backslash le\; p$, since $H\_j/G\text{'}$ is cyclic of order $p^2$, whereas $H\_\{p+1\}/G\text{'}$ is bicyclic of type $(p,p)$.

For each $1\backslash le\; i\backslash le\; p+1$, let $T\_\{2,i\}:\; G\backslash to\; U\_i/U\_i\text{'}$ be the Artin transfer homomorphism from $G$ to the abelianization of $U\_i$.

Definition. The family $\backslash varkappa\_\{2,U,H\}(G)=(\backslash ker(T\_\{2,i\}))\_\{i=1\}^\{p+1\}$ is called the second layer transfer kernel type of $G$ with respect to $U\_1,\backslash ldots,U\_\{p+1\}$ and $H\_1,\backslash ldots,H\_\{p+1\}$, and is identified with $(\backslash varkappa\_2(i))\_\{i=1\}^\{p+1\},$ where

:

Combining the information on the two layers, we obtain the (complete) transfer kernel type $\backslash varkappa\_\{H,U\}(G)=(\backslash varkappa\_\{1,H,U\}(G);\backslash varkappa\_\{2,U,H\}(G))$ of the p-group $G$ with respect to $H\_1,\backslash ldots,H\_\{p+1\}$ and $U\_1,\backslash ldots,U\_\{p+1\}$.

Remark. The distinguished subgroups $H\_\{p+1\}$ and $U\_\{p+1\}=\backslash Phi(G)$ are unique invariants of $G$ and should not be renumerated. However, independent renumerations of the remaining maximal subgroups $K\_i=H\_\{\backslash tau(i)\}\; (1\backslash le\; i\backslash le\; p)$ and the transfers $V\_\{1,i\}=T\_\{1,\backslash tau(i)\}$ by means of a permutation $\backslash tau\backslash in\; S\_p$, and of the remaining subgroups $W\_i=U\_\{\backslash sigma(i)\}\; (1\backslash le\; i\backslash le\; p)$ of index $p^2$ and the transfers $V\_\{2,i\}=T\_\{2,\backslash sigma(i)\}$ by means of a permutation $\backslash sigma\backslash in\; S\_p$, give rise to new TKTs $\backslash lambda\_\{1,K,W\}(G)=(\backslash ker(V\_\{1,i\}))\_\{i=1\}^\{p+1\}$ with respect to $K\_1,\backslash ldots,K\_\{p+1\}$ and $W\_1,\; \backslash ldots,\; W\_\{p+1\}$, identified with $(\backslash lambda\_1(i))\_\{i=1\}^\{p+1\}$, where

:

and $\backslash lambda\_\{2,W,K\}(G)=(\backslash ker(V\_\{2,i\}))\_\{i=1\}^\{p+1\}$ with respect to $W\_1,\backslash ldots,W\_\{p+1\}$ and $K\_1,\backslash ldots,K\_\{p+1\}$, identified with $(\backslash lambda\_2(i))\_\{i=1\}^\{p+1\},$ where

:

It is adequate to view the TKTs $\backslash lambda\_\{1,K,W\}(G)\backslash sim\backslash varkappa\_\{1,H,U\}(G)$ and $\backslash lambda\_\{2,W,K\}(G)\backslash sim\; \backslash varkappa\_\{2,U,H\}\; (G)$ as equivalent. Since we have

:$\backslash begin\{align\}$

W_{\lambda_1(i)} &=\ker(V_{1,i})=\ker(T_{1,\hat{\tau}(i)})=U_{\varkappa_1(\hat{\tau}(i))}=W_{\tilde{\sigma}^{-1}(\varkappa_1(\hat{\tau}(i)))} \\

K_{\lambda_2(i)} &=\ker(V_{2,i})=\ker(T_{2,\hat{\sigma}(i)})=H_{\varkappa_2(\hat{\sigma}(i))}=K_{\tilde{\tau}^{-1}(\varkappa_2(\hat{\sigma}(i)))}

\end{align}

the relations between $\backslash lambda\_1$ and $\backslash varkappa\_1$, and $\backslash lambda\_2$ and $\backslash varkappa\_2$, are given by

:$\backslash lambda\_1=\backslash tilde\{\backslash sigma\}^\{-1\}\backslash circ\backslash varkappa\_1\backslash circ\backslash hat\{\backslash tau\}$

:$\backslash lambda\_2=\backslash tilde\{\backslash tau\}^\{-1\}\; \backslash circ\backslash varkappa\_2\backslash circ\backslash hat\{\backslash sigma\}$

Therefore, $\backslash lambda=(\backslash lambda\_1,\backslash lambda\_2)$ is another representative of the orbit $\backslash varkappa^\{S\_p\backslash times\; S\_p\}$ of $\backslash varkappa=(\backslash varkappa\_1,\backslash varkappa\_2)$ under the action:

:$((\backslash sigma,\backslash tau),(\backslash mu\_1,\backslash mu\_2))\backslash mapsto\; \backslash left\; (\backslash tilde\{\backslash sigma\}^\{-1\}\backslash circ\backslash mu\_1\backslash circ\backslash hat\backslash tau,\; \backslash tilde\{\backslash tau\}^\{-1\}\backslash circ\; \backslash mu\_2\backslash circ\; \backslash hat\backslash sigma\; \backslash right\; )$

of the product of two symmetric groups $S\_p\backslash times\; S\_p$ on the set of all pairs of mappings $\backslash \{\; 1,\backslash ldots,p+1\backslash \}\; \backslash to\; \backslash \{\; 0,1,\; \backslash ldots,p+1\backslash \}$, where the extensions $\backslash hat\{\backslash pi\}\backslash in\; S\_\{p+1\}$ and $\backslash tilde\{\backslash pi\}\backslash in\; S\_\{p+2\}$ of a permutation $\backslash pi\backslash in\; S\_p$ are defined by $\backslash hat\{\backslash pi\}(p+1)=\backslash tilde\{\backslash pi\}(p+1)=p+1$ and $\backslash tilde\{\backslash pi\}(0)=0$, and formally $H\_0=K\_0=G,\; K\_\{p+1\}=H\_\{p+1\},\; U\_0=W\_0=H\_\{p+1\},$ and $W\_\{p+1\}=U\_\{p+1\}=\backslash Phi(G).$

Definition. The orbit $\backslash varkappa(G)=\backslash varkappa^\{S\_p\backslash times\; S\_p\}$ of any representative $\backslash varkappa=(\backslash varkappa\_1,\; \backslash varkappa\_2)$ is an invariant of the p-group $G$ and is called its transfer kernel type, briefly TKT.

The Artin transfer $T\_\{2,i\}:\; G\backslash to\; U\_i/U\_i\text{'}$ is the composition $T\_\{2,i\}=\backslash tilde\{T\}\_\{H\_j,U\_i\}\backslash circ\; T\_\{1,j\}$ of the induced transfer $\backslash tilde\{T\}\_\{H\_j,U\_i\}:\; H\_j/H\_j\text{'}\backslash to\; U\_i/U\_i\text{'}$ from $H\_j$ to $U\_i$ and the Artin transfer $T\_\{1,j\}:\; G\backslash to\; H\_j/H\_j\text{'}.$

There are two options regarding the intermediate subgroups

• For the subgroups $U\_1,\backslash ldots,U\_p$ only the distinguished maximal subgroup $H\_\{p+1\}$ is an intermediate subgroup.
• For the Frattini subgroup $U\_\{p+1\}=\backslash Phi(G)$ all maximal subgroups $H\_1,\backslash ldots,H\_\{p+1\}$ are intermediate subgroups.

:This causes restrictions for the transfer kernel type $\backslash varkappa\_2(G)$ of the second layer, since

::$\backslash ker(T\_\{2,i\})=\backslash ker(\backslash tilde\{T\}\_\{H\_j,U\_i\}\backslash circ\; T\_\{1,j\})\backslash supset\backslash ker(T\_\{1,j\}),$

:and thus

::$\backslash forall\; i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; p\backslash \}:\; \backslash qquad\; \backslash ker(T\_\{2,i\})\backslash supset\backslash ker(T\_\{1,p+1\}).$

:But even

::$\backslash ker(T\_\{2,p+1\})\backslash supset\; \backslash left\; \backslash langle\backslash bigcup\_\{j=1\}^\{p+1\}\; \backslash ker(T\_\{1,j\})\; \backslash right\; \backslash rangle.$

:Furthermore, when $G=\backslash langle\; x,y\backslash rangle$ with $x^p\backslash notin\; G\text{'},\; y^p\backslash in\; G\text{'},$ an element $xy^\{k-1\}\; (1\backslash le\; k\backslash le\; p)$ of order $p^2$ with respect to $G\text{'}$, can belong to $\backslash ker(T\_\{2,i\})$ only if its $p$th power is contained in $\backslash ker(T\_\{1,j\})$, for all intermediate subgroups

File:FactorThruDerivedQuotient.png

The common feature of all parent-descendant relations between finite p-groups is that the parent $\backslash pi(G)$ is a quotient $G/N$ of the descendant $G$ by a suitable normal subgroup $N.$ Thus, an equivalent definition can be given by selecting an epimorphism $\backslash phi:\; G\; \backslash to\; \backslash tilde\{G\}$ with $\backslash ker(\backslash phi)\; =\; N.$ Then the group $\backslash tilde\{G\}=\backslash phi(G)$ can be viewed as the parent of the descendant $G$.

In the following sections, this point of view will be taken, generally for arbitrary groups, not only for finite p-groups.

:Proposition. Suppose $A$ is an abelian group and $\backslash phi:\; G\backslash to\; A$ is a homomorphism. Let $\backslash omega:\; G\backslash to\; G/G\text{'}$ denote the canonical projection map. Then there exists a unique homomorphism $\backslash tilde\{\backslash phi\}:\; G/G\text{'}\backslash to\; A$ such that $\backslash phi=\backslash tilde\{\backslash phi\}\backslash circ\backslash omega$ and $\backslash ker(\backslash tilde\{\backslash phi\})=\backslash ker(\backslash phi)/G\text{'}$ (See Figure 1).

Proof. This statement is a consequence of the second Corollary in the article on the induced homomorphism. Nevertheless, we give an independent proof for the present situation: the uniqueness of $\backslash tilde\{\backslash phi\}$ is a consequence of the condition $\backslash phi=\; \backslash tilde\{\backslash phi\}\backslash circ\backslash omega,$ which implies for any $x\backslash in\; G$ we have:

:$\backslash tilde\{\backslash phi\}(xG\text{'})=\backslash tilde\{\backslash phi\}(\backslash omega(x))=(\backslash tilde\{\backslash phi\}\backslash circ\backslash omega)(x)=\backslash phi(x),$

$\backslash tilde\{\backslash phi\}$ is a homomorphism, let $x,y\backslash in\; G$ be arbitrary, then:

:$\backslash begin\{align\}$

\tilde{\phi} \left (xG'\cdot yG' \right ) &= \tilde{\phi}((xy)G')=\phi(xy)=\phi(x)\cdot\phi(y)=\tilde{\phi}(xG') \cdot \tilde{\phi}(xG') \\

\phi([x,y]) &=\phi \left (x^{-1}y^{-1}xy \right )=\phi(x^{-1})\phi(y^{-1})\phi(x)\phi(y)=[\phi(x),\phi(y)]=1 && A \text{ is abelian.}

\end{align}

Thus, the commutator subgroup $G\text{'}\; \backslash subset\; \backslash ker(\backslash phi)$, and this finally shows that the definition of $\backslash tilde\{\backslash phi\}$ is independent of the coset representative,

:$\backslash begin\{align\}$

xG'=yG' &\Longrightarrow y^{-1}x\in G'\subset \ker(\phi) \\

&\Longrightarrow 1 = \phi(y^{-1}x) = \tilde{\phi}(y^{-1}xG')= \tilde{\phi}(yG')^{-1}\cdot\tilde{\phi}(xG')\\

&\Longrightarrow \tilde{\phi}(xG')=\tilde{\phi}(yG')

\end{align}

File:EpiAndDerivedQuotients.png

:Proposition. Assume $G,\; \backslash tilde\{G\},\; \backslash phi$ are as above and $\backslash tilde\{H\}=\backslash phi(H)$ is the image of a subgroup $H.$ The commutator subgroup of $\backslash tilde\{H\}$ is the image of the commutator subgroup of $H.$ Therefore, $\backslash phi$ induces a unique epimorphism $\backslash tilde\{\backslash phi\}:\; H/H\text{'}\; \backslash to\; \backslash tilde\{H\}/\; \backslash tilde\{H\}\text{'}$, and thus $\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}$ is a quotient of $H/H\text{'}.$ Moreover, if $\backslash ker(\backslash phi)\backslash le\; H\text{'}$, then the map $\backslash tilde\{\backslash phi\}$ is an isomorphism (See Figure 2).

Proof. This claim is a consequence of the Main Theorem in the article on the induced homomorphism. Nevertheless, an independent proof is given as follows: first, the image of the commutator subgroup is

:$\backslash phi(H\text{'})=\backslash phi([\; H,H])=\backslash phi(\backslash langle[\; u,v]\; |\; u,v\backslash in\; H\backslash rangle)=\backslash langle[\; \backslash phi(u),\backslash phi(v)]\backslash mid\; u,v\backslash in\; H\backslash rangle=[\; \backslash phi(H),\backslash phi(H)]=\backslash phi(H)\text{'}=\backslash tilde\{H\}\text{'}.$

Second, the epimorphism $\backslash phi$ can be restricted to an epimorphism $\backslash phi|\_H:\; H\backslash to\backslash tilde\{H\}$. According to the previous section, the composite epimorphism $(\backslash omega\_\{\backslash tilde\{H\}\}\backslash circ\backslash phi|\_H):\; H\backslash to\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}$ factors through $H/H\text{'}$ by means of a uniquely determined epimorphism $\backslash tilde\{\backslash phi\}:\; H/H\text{'}\backslash to\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}$ such that $\backslash tilde\{\backslash phi\}\backslash circ\backslash omega\_H=\backslash omega\_\{\backslash tilde\{H\}\}\backslash circ\backslash phi|\_H$. Consequently, we have $\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}\backslash simeq\; (H/H\text{'})/\backslash ker(\backslash tilde\{\backslash phi\})$. Furthermore, the kernel of $\backslash tilde\{\backslash phi\}$ is given explicitly by $\backslash ker(\backslash tilde\{\backslash phi\})=(H\text{'}\backslash cdot\backslash ker(\backslash phi))/H\text{'}$.

Finally, if $\backslash ker(\backslash phi)\backslash le\; H\text{'}$, then $\backslash tilde\{\backslash phi\}$ is an isomorphism, since $\backslash ker\; (\backslash tilde\{\backslash phi\})\; =H\text{'}/H\text{'}=1$.

Definition.{{cite journal|author=Bush, M. R. |author2=Mayer, D. C.|year=2015| title=3-class field towers of exact length 3| journal=J. Number Theory| volume=147| pages=766–777 (preprint: arXiv:1312.0251 [math.NT], 2013)| doi=10.1016/j.jnt.2014.08.010| arxiv=1312.0251|s2cid=119147524}} Due to the results in the present section, it makes sense to define a partial order on the set of abelian type invariants by putting $\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}\backslash preceq\; H/H\text{'}$, when $\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}\backslash simeq\; (H/H\text{'})/\backslash ker(\backslash tilde\{\backslash phi\})$, and $\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}=H/H\text{'}$, when $\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}\backslash simeq\; H/H\text{'}$.

File:EpiAndArtinTransfers.png

:Proposition. Assume $G,\; \backslash tilde\{G\},\; \backslash phi$ are as above and $\backslash tilde\{H\}=\backslash phi(H)$ is the image of a subgroup of finite index $n.$ Let $T\_\{G,H\}:\; G\; \backslash to\; H/H\text{'}$ and $T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}\; :\; \backslash tilde\{G\}\; \backslash to\; \backslash tilde\{H\}/\backslash tilde\{H\}\text{'}$ be Artin transfers. If $\backslash ker(\backslash phi)\backslash le\; H$, then the image of a left transversal of $H$ in $G$ is a left transversal of $\backslash tilde\{H\}$ in $\backslash tilde\{G\}$, and $\backslash phi(\backslash ker(T\_\{G,H\}))\; \backslash subset\; \backslash ker(T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}).$ Moreover, if $\backslash ker(\backslash phi)\backslash le\; H\text{'}$ then $\backslash phi(\backslash ker(T\_\{G,H\}))\; =\; \backslash ker\; (T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\})$ (See Figure 3).

Proof. Let $(g\_1,\backslash ldots,g\_n)$ be a left transversal of $H$ in $G$. Then we have a disjoint union:

:$G=\backslash bigsqcup\_\{i=1\}^n\; g\_iH.$

Consider the image of this disjoint union, which is not necessarily disjoint,

:$\backslash phi(G)=\backslash bigcup\_\{i=1\}^n\; \backslash phi(g\_i)\backslash phi(H),$

and let $j,k\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}.$ We have:

:$\backslash begin\{align\}$

\phi(g_j)\phi(H)=\phi(g_k)\phi(H) &\Longleftrightarrow \phi(H)=\phi(g_j)^{-1} \phi(g_k)\phi(H)= \phi(g_j^{-1}g_k)\phi(H) \\

&\Longleftrightarrow \phi(g_j^{-1}g_k)=\phi(h) && \text{for some } h \in H \\

&\Longleftrightarrow \phi(h^{-1}g_j^{-1}g_k)=1 \\

&\Longleftrightarrow h^{-1}g_j^{-1}g_k \in \ker(\phi) \subset H \\

&\Longleftrightarrow g_j^{-1}g_k \in H \\

&\Longleftrightarrow j=k \\

\end{align}

Let $\backslash tilde\{\backslash phi\}:\; H/H\text{'}\backslash to\backslash tilde\{H\}/\backslash tilde\{H\}\text{'}$ be the epimorphism from the previous proposition. We have:

:$\backslash tilde\{\backslash phi\}(T\_\{G,H\}(x))=\backslash tilde\{\backslash phi\}\; \backslash left\; (\backslash prod\_\{i=1\}^n\; g\_\{\backslash pi\_x(i)\}^\{-1\}xg\_i\backslash cdot\; H\text{'}\; \backslash right\; )=\backslash prod\_\{i=1\}^n\; \backslash phi\; \backslash left\; (g\_\{\backslash pi\_x(i)\}\; \backslash right\; )^\{-1\}\backslash phi(x)\backslash phi(g\_i)\backslash cdot\backslash phi(H\text{'}).$

Since $\backslash phi(H\text{'})=\backslash phi(H)\text{'}=\backslash tilde\{H\}\text{'}$, the right hand side equals $T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}(\backslash phi(x))$, if $(\backslash phi(g\_1),\backslash ldots,\backslash phi(g\_n))$ is a left transversal of $\backslash tilde\{H\}$ in $\backslash tilde\{G\}$, which is true when $\backslash ker(\backslash phi)\backslash subset\; H.$ Therefore, $\backslash tilde\{\backslash phi\}\backslash circ\; T\_\{G,H\}=T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}\backslash circ\backslash phi.$ Consequently, $\backslash ker(\backslash phi)\backslash subset\; H$ implies the inclusion

:$\backslash phi(\backslash ker(T\_\{G,H\}))\backslash subset\; \backslash ker(T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}).$

Finally, if $\backslash ker(\backslash phi)\backslash subset\; H\text{'}$, then by the previous proposition $\backslash tilde\{\backslash phi\}$ is an isomorphism. Using its inverse we get $T\_\{G,H\}=\backslash tilde\{\backslash phi\}^\{-1\}\backslash circ\; T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}\backslash circ\backslash phi$, which proves

:$\backslash phi^\{-1\}\; \backslash left\; (\backslash ker(T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\})\; \backslash right\; )\backslash subset\; \backslash ker(T\_\{G,H\}).$

Combining the inclusions we have:

:$\backslash begin\{align\}$

\begin{cases} \phi(\ker(T_{G,H}))\subset \ker(T_{\tilde{G},\tilde{H}}) \\ \phi^{-1} \left (\ker(T_{\tilde{G},\tilde{H}}) \right )\subset \ker(T_{G,H})\end{cases} &\Longrightarrow \begin{cases} \phi(\ker(T_{G,H}))\subset \ker(T_{\tilde{G},\tilde{H}}) \\ \phi \left (\phi^{-1} \left (\ker(T_{\tilde{G},\tilde{H}}) \right ) \right )\subset \phi(\ker(T_{G,H}))\end{cases} \\[8pt]

&\Longrightarrow \phi \left (\phi^{-1} \left (\ker(T_{\tilde{G},\tilde{H}}) \right ) \right )\subset \phi(\ker(T_{G,H})) \subset \ker(T_{\tilde{G},\tilde{H}}) \\[8pt]

&\Longrightarrow \ker(T_{\tilde{G},\tilde{H}}) \subset \phi(\ker(T_{G,H})) \subset \ker(T_{\tilde{G},\tilde{H}}) \\[8pt]

&\Longrightarrow \phi(\ker(T_{G,H}))=\ker(T_{\tilde{G},\tilde{H}})

\end{align}

Definition. In view of the results in the present section, we are able to define a partial order of transfer kernels by setting $\backslash ker(T\_\{G,H\})\backslash preceq\backslash ker(T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\})$, when $\backslash phi(\backslash ker(T\_\{G,H\}))\; \backslash subset\; \backslash ker(T\_\{\backslash tilde\{G\},\backslash tilde\{H\}\}).$

Assume $G,\; \backslash tilde\{G\},\; \backslash phi$ are as above and that $G/G\text{'}$ and $\backslash tilde\{G\}/\backslash tilde\{G\}\text{'}$ are isomorphic and finite. Let $(H\_i)\_\{i\backslash in\; I\}$ denote the family of all subgroups containing $G\text{'}$ (making it a finite family of normal subgroups). For each $i\backslash in\; I$ let:

:$\backslash begin\{align\}$

\tilde{H_i} &:= \phi (H_i) \\

T_i &:= T_{G,H_i}: G \to H_i/H_i' \\

\tilde{T_i} &:= T_{\tilde{G}, \tilde{H_i}}: \tilde{G} \to \tilde{H_i} /\tilde{H_i}'

\end{align}

Take $J$ be any non-empty subset of $I$. Then it is convenient to define $\backslash varkappa\_H(G)=(\backslash ker(T\_j))\_\{j\backslash in\; J\}$, called the (partial) transfer kernel type (TKT) of $G$ with respect to $(H\_j)\_\{j\backslash in\; J\}$, and $\backslash tau\_H(G)=(H\_j/H\_j\text{'})\_\{j\backslash in\; J\},$ called the (partial) transfer target type (TTT) of $G$ with respect to $(H\_j)\_\{j\backslash in\; J\}$.

Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey the following fundamental inheritance laws:

:Inheritance Law I. If $\backslash ker(\backslash phi)\backslash le\; \backslash cap\_\{j\backslash in\; J\}\; H\_j$, then $\backslash tau\_\{\backslash tilde\{H\}\}(\backslash tilde\{G\})\backslash preceq\backslash tau\_H(G)$, in the sense that $\backslash tilde\{H\_j\}/\backslash tilde\{H\_j\}\text{'}\backslash preceq\; H\_j/H\_j\text{'}$, for each $j\backslash in\; J$, and $\backslash varkappa\_H(G)\; \backslash preceq\; \backslash varkappa\_\{\backslash tilde\{H\}\}\; (\backslash tilde\{G\})$, in the sense that $\backslash ker(T\_j)\backslash preceq\backslash ker(\backslash tilde\{T\_j\})$, for each $j\backslash in\; J$.

:Inheritance Law II. If $\backslash ker(\backslash phi)\backslash le\; \backslash cap\_\{j\backslash in\; J\}\; H\_j\text{'}$, then $\backslash tau\_\{\backslash tilde\{H\}\}(\backslash tilde\{G\})=\backslash tau\_H(G)$, in the sense that $\backslash tilde\{H\_j\}/\backslash tilde\{H\_j\}\text{'}=H\_j/H\_j\text{'}$, for each $j\backslash in\; J$, and $\backslash varkappa\_H(G)=\backslash varkappa\_\{\backslash tilde\{H\}\}(\backslash tilde\{G\})$, in the sense that $\backslash ker(T\_j)=\backslash ker(\backslash tilde\{T\_j\})$, for each $j\backslash in\; J$.

A further inheritance property does not immediately concern Artin transfers but will prove to be useful in applications to descendant trees.

:Inheritance Law III. Assume $G,\; \backslash tilde\{G\},\; \backslash phi$ are as above and $\backslash sigma\backslash in\backslash mathrm\{Aut\}(G).$ If $\backslash sigma\; (\backslash ker(\backslash phi))\; \backslash subset\backslash ker(\backslash phi)$ then there exists a unique epimorphism $\backslash tilde\{\backslash sigma\}:\; \backslash tilde\{G\}\; \backslash to\; \backslash tilde\{G\}$ such that $\backslash phi\backslash circ\backslash sigma=\backslash tilde\{\backslash sigma\}\backslash circ\backslash phi$. If $\backslash sigma\; (\backslash ker(\backslash phi))=\; \backslash ker(\backslash phi),$ then $\backslash tilde\{\backslash sigma\}\; \backslash in\backslash mathrm\{Aut\}(\backslash tilde\{G\}).$

Proof. Using the isomorphism $\backslash tilde\{G\}=\backslash phi(G)\backslash simeq\; G/\backslash ker(\backslash phi)$ we define:

:$\backslash begin\{cases\}\; \backslash tilde\{\backslash sigma\}:\; \backslash tilde\{G\}\; \backslash to\; \backslash tilde\{G\}\; \backslash \backslash \; \backslash tilde\{\backslash sigma\}(g\backslash ker(\backslash phi))\; :=\; \backslash sigma(g)\; \backslash ker(\backslash phi)\; \backslash end\{cases\}$

First we show this map is well-defined:

:$\backslash begin\{align\}$

g\ker(\phi)=h\ker(\phi) &\Longrightarrow h^{-1}g\in\ker(\phi) \\

&\Longrightarrow \sigma(h^{-1}g)\in \sigma (\ker(\phi)) \\

&\Longrightarrow \sigma(h^{-1}g)\in \ker(\phi) && \sigma (\ker(\phi)) \subset \ker(\phi) \\

&\Longrightarrow \sigma(h^{-1})\sigma(g)\in \ker(\phi) \\

&\Longrightarrow \sigma(g) \ker(\phi) =\sigma(h)\ker(\phi)

\end{align}

The fact that $\backslash tilde\{\backslash sigma\}$ is surjective, a homomorphism and satisfies $\backslash phi\backslash circ\backslash sigma=\backslash tilde\{\backslash sigma\}\backslash circ\backslash phi$ are easily verified.

And if $\backslash sigma(\backslash ker(\backslash phi))=\backslash ker(\backslash phi)$, then injectivity of $\backslash tilde\{\backslash sigma\}$ is a consequence of

:$\backslash begin\{align\}$

\tilde{\sigma}(g\ker(\phi)) = \ker(\phi) &\Longrightarrow \sigma(g)\ker(\phi)=\ker(\phi) \\

&\Longrightarrow \sigma(g)\in\ker(\phi) \\

&\Longrightarrow \sigma^{-1}(\sigma(g))\in \sigma^{-1}(\ker(\phi)) \\

&\Longrightarrow g \in \sigma^{-1}(\ker(\phi)) \\

&\Longrightarrow g \in \ker(\phi) && \sigma^{-1}(\ker(\phi))\subset \ker(\phi) \\

&\Longrightarrow g \ker(\phi) = \ker(\phi)

\end{align}

Let $\backslash omega:\; G\backslash to\; G/G\text{'}$ be the canonical projection then there exists a unique induced automorphism $\backslash bar\{\backslash sigma\}\backslash in\backslash mathrm\{Aut\}(G/G\text{'})$ such that $\backslash omega\backslash circ\backslash sigma\; =\backslash bar\{\backslash sigma\}\; \backslash circ\; \backslash omega$, that is,

:$\backslash forall\; g\; \backslash in\; G:\; \backslash qquad\; \backslash bar\{\backslash sigma\}(gG\text{'})=\backslash bar\{\backslash sigma\}(\backslash omega(g))=\backslash omega(\backslash sigma(g))=\backslash sigma(g)G\text{'},$

The reason for the injectivity of $\backslash bar\{\backslash sigma\}$ is that

:$\backslash sigma(g)G\text{'}=\backslash bar\{\backslash sigma\}(gG\text{'})=G\text{'}\; \backslash Rightarrow\; \backslash sigma(g)\backslash in\; G\text{'}\; \backslash Rightarrow\; g=\backslash sigma^\{-1\}(\backslash sigma(g))\backslash in\; G\text{'},$

since $G\text{'}$ is a characteristic subgroup of $G$.

Definition. $G$ is called a σ−group, if there exists $\backslash sigma\backslash in\backslash mathrm\{Aut\}(G)$ such that the induced automorphism acts like the inversion on $G/G\text{'}$, that is for all

:$g\; \backslash in\; G:\; \backslash qquad\; \backslash sigma(g)G\text{'}=\backslash bar\{\backslash sigma\}(gG\text{'})=g^\{-1\}G\text{'}\; \backslash Longleftrightarrow\; \backslash sigma(g)g\backslash in\; G\text{'}.$

The Inheritance Law III asserts that, if $G$ is a σ−group and $\backslash sigma\; (\backslash ker(\backslash phi))\; =\backslash ker(\backslash phi)$, then $\backslash tilde\{G\}$ is also a σ−group, the required automorphism being $\backslash tilde\{\backslash sigma\}$. This can be seen by applying the epimorphism $\backslash phi$ to the equation $\backslash sigma(g)G\text{'}=\backslash bar\{\backslash sigma\}(gG\text{'})=g^\{-1\}G\text{'}$ which yields

:$\backslash forall\; x=\backslash phi(g)\backslash in\backslash phi(G)=\backslash tilde\{G\}:\; \backslash qquad\; \backslash tilde\{\backslash sigma\}(x)\backslash tilde\{G\}\text{'}=\backslash tilde\{\backslash sigma\}(\backslash phi(g))\backslash tilde\{G\}\text{'}\; =\backslash phi\; (\backslash sigma(g))\backslash phi(G\text{'})\; =\backslash phi(g^\{-1\})\; \backslash phi(G\text{'})=\; \backslash phi(g)^\{-1\}\backslash tilde\{G\}\text{'}=x^\{-1\}\backslash tilde\{G\}\text{'}.$

In this section, the results concerning the inheritance of TTTs and TKTs from quotients in the previous section are applied to the simplest case, which is characterized by the following

Assumption. The parent $\backslash pi(G)$ of a group $G$ is the quotient $\backslash pi(G)=G/N$ of $G$ by the last non-trivial term $N=\backslash gamma\_c(G)\backslash triangleleft\; G$ of the lower central series of $G$, where $c$ denotes the nilpotency class of $G$. The corresponding epimorphism $\backslash pi$ from $G$ onto $\backslash pi(G)=G/\backslash gamma\_c(G)$ is the canonical projection, whose kernel is given by $\backslash ker(\backslash pi)=\backslash gamma\_c(G)$.

Under this assumption,

the kernels and targets of Artin transfers turn out to be compatible with parent-descendant relations between finite p-groups.

Compatibility criterion. Let $p$ be a prime number. Suppose that $G$ is a non-abelian finite p-group of nilpotency class $c=\backslash mathrm\{cl\}(G)\backslash ge\; 2$. Then the TTT and the TKT of $G$ and of its parent $\backslash pi(G)$ are comparable in the sense that $\backslash tau(\backslash pi(G))\backslash preceq\backslash tau(G)$ and $\backslash varkappa(G)\backslash preceq\backslash varkappa(\backslash pi(G))$.

The simple reason for this fact is that, for any subgroup $G\text{'}\backslash le\; H\backslash le\; G$, we have $\backslash ker(\backslash pi)=\backslash gamma\_c(G)\; \backslash le\backslash gamma\_2(G)=G\text{'}\backslash le\; H$, since $c\backslash ge\; 2$.

For the remaining part of this section, the investigated groups are supposed to be finite metabelian p-groups $G$ with elementary abelianization $G/G\text{'}$ of rank $2$, that is of type $(p,p)$.

Partial stabilization for maximal class. A metabelian p-group $G$ of coclass $\backslash mathrm\{cc\}(G)=1$ and of nilpotency class $c=\backslash mathrm\{cl\}(G)\backslash ge\; 3$ shares the last $p$ components of the TTT $\backslash tau(G)$ and of the TKT $\backslash varkappa(G)$ with its parent $\backslash pi(G)$. More explicitly, for odd primes $p\backslash ge\; 3$, we have $\backslash tau(G)\_i=(p,p)$ and $\backslash varkappa(G)\_i=0$ for $2\backslash le\; i\backslash le\; p+1$.

{{cite journal|

author=Mayer, D. C.|

year=2012|

title=Transfers of metabelian p-groups|

journal=Monatsh. Math.|

volume=166|

number=3–4|

pages=467–495|

doi=10.1007/s00605-010-0277-x|

arxiv=1403.3896|

s2cid=119167919}}

This criterion is due to the fact that $c\backslash ge\; 3$ implies $\backslash ker(\backslash pi)=\backslash gamma\_c(G)\backslash le\backslash gamma\_3(G)=H\_i\text{'}$,

{{cite journal|

author=Mayer, D. C.|

year=2012|

title=The second p-class group of a number field|

journal=Int. J. Number Theory|

volume=8|

number=2|

pages=471–505

|doi=10.1142/s179304211250025x|

arxiv=1403.3899|

s2cid=119332361}}

for the last $p$ maximal subgroups $H\_2,\backslash ldots,H\_\{p+1\}$ of $G$.

The condition $c\backslash ge\; 3$ is indeed necessary for the partial stabilization criterion. For odd primes $p\backslash ge\; 3$, the extra special $p$-group $G=G^3\_0(0,1)$ of order $p^3$ and exponent $p^2$ has nilpotency class $c=2$ only, and the last $p$ components of its TKT $\backslash varkappa=(1^\{p+1\})$ are strictly smaller than the corresponding components of the TKT $\backslash varkappa=(0^\{p+1\})$ of its parent $\backslash pi(G)$ which is the elementary abelian $p$-group of type $(p,p)$.

For $p=2$, both extra special $2$-groups of coclass $1$ and class $c=2$, the ordinary quaternion group $G=G^3\_0(0,1)$ with TKT $\backslash varkappa=(123)$ and the dihedral group $G=G^3\_0(0,0)$ with TKT $\backslash varkappa=(023)$, have strictly smaller last two components of their TKTs than their common parent $\backslash pi(G)=C\_2\backslash times\; C\_2$ with TKT $\backslash varkappa=(000)$.

Total stabilization for maximal class and positive defect.

A metabelian p-group $G$ of coclass $\backslash mathrm\{cc\}(G)=1$ and of nilpotency class $c=m-1=\backslash mathrm\{cl\}(G)\backslash ge\; 4$, that is, with index of nilpotency $m\backslash ge\; 5$, shares all $p+1$ components of the TTT $\backslash tau(G)$ and of the TKT $\backslash varkappa(G)$ with its parent $\backslash pi(G)$, provided it has positive defect of commutativity $k=k(G)\backslash ge\; 1$.

Note that $k\backslash ge\; 1$ implies $p\backslash ge\; 3$, and we have $\backslash varkappa(G)\_i=0$ for all $1\backslash le\; i\backslash le\; p+1$.

This statement can be seen by observing that the conditions $m\backslash ge\; 5$ and $k\backslash ge\; 1$ imply $\backslash ker(\backslash pi)=\backslash gamma\_\{m-1\}(G)\backslash le\backslash gamma\_\{m-k\}(G)\backslash le\; H\_i\text{'}$,

for all the $p+1$ maximal subgroups $H\_1,\backslash ldots,H\_\{p+1\}$ of $G$.

The condition $k\backslash ge\; 1$ is indeed necessary for total stabilization. To see this it suffices to consider the first component of the TKT only. For each nilpotency class $c\backslash ge\; 4$, there exist (at least) two groups $G=G^\{c+1\}\_0(0,1)$ with TKT $\backslash varkappa=(10^p)$ and $G=G^\{c+1\}\_0(1,0)$ with TKT $\backslash varkappa=(20^p)$, both with defect $k=0$, where the first component of their TKT is strictly smaller than the first component of the TKT $\backslash varkappa=(0^\{p+1\})$ of their common parent $\backslash pi(G)=G^c\_0(0,0)$.

Partial stabilization for non-maximal class.

Let $p=3$ be fixed. A metabelian 3-group $G$ with abelianization $G/G\text{'}\backslash simeq\; (3,3)$, coclass $\backslash mathrm\{cc\}(G)\backslash ge\; 2$ and nilpotency class $c=\backslash mathrm\{cl\}(G)\backslash ge\; 4$ shares the last two (among the four) components of the TTT $\backslash tau(G)$ and of the TKT $\backslash varkappa(G)$ with its parent $\backslash pi(G)$.

This criterion is justified by the following consideration. If $c\backslash ge\; 4$, then $\backslash ker(\backslash pi)=\backslash gamma\_c(G)\backslash le\backslash gamma\_4(G)\backslash le\; H\_i\text{'}$

for the last two maximal subgroups $H\_3,H\_4$ of $G$.

The condition $c\backslash ge\; 4$ is indeed unavoidable for partial stabilization, since there exist several $3$-groups of class $c=3$, for instance those with SmallGroups identifiers $G\backslash in\backslash \{\backslash langle\; 243,3\backslash rangle,\backslash langle\; 243,6\backslash rangle,\backslash langle\; 243,8\backslash rangle\backslash \}$, such that the last two components of their TKTs $\backslash varkappa\backslash in\backslash \{\; (0043),(0122),(2034)\backslash \}$ are strictly smaller than the last two components of the TKT $\backslash varkappa=(0000)$ of their common parent $\backslash pi(G)=G^3\_0(0,0)$.

Total stabilization for non-maximal class and cyclic centre.

Again, let $p=3$ be fixed.

A metabelian 3-group $G$ with abelianization $G/G\text{'}\backslash simeq\; (3,3)$, coclass $\backslash mathrm\{cc\}(G)\backslash ge\; 2$, nilpotency class $c=\backslash mathrm\{cl\}(G)\backslash ge\; 4$ and cyclic centre $\backslash zeta\_1(G)$ shares all four components of the TTT $\backslash tau(G)$ and of the TKT $\backslash varkappa(G)$ with its parent $\backslash pi(G)$.

The reason is that, due to the cyclic centre, we have $\backslash ker(\backslash pi)=\backslash gamma\_c(G)=\backslash zeta\_1(G)\backslash le\; H\_i\text{'}$

for all four maximal subgroups $H\_1,\backslash ldots,H\_4$ of $G$.

The condition of a cyclic centre is indeed necessary for total stabilization, since for a group with bicyclic centre there occur two possibilities.

Either $\backslash gamma\_c(G)=\backslash zeta\_1(G)$ is also bicyclic, whence $\backslash gamma\_c(G)$ is never contained in $H\_2\text{'}$,

or is cyclic but is never contained in $H\_1\text{'}$.

Summarizing, we can say that the last four criteria underpin the fact that Artin transfers provide a marvellous tool for classifying finite p-groups.

In the following sections, it will be shown how these ideas can be applied for endowing descendant trees with additional structure, and for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in pure group theory and in algebraic number theory.

File:TreeCoclass2RootQ.png

This section uses the terminology of descendant trees in the theory of finite p-groups.

In Figure 4, a descendant tree with modest complexity is selected exemplarily to demonstrate how Artin transfers provide additional structure for each vertex of the tree.

More precisely, the underlying prime is $p=3$, and the chosen descendant tree is actually a coclass tree having a unique infinite mainline, branches of depth $3$, and strict periodicity of length $2$ setting in with branch $\backslash mathcal\{B\}(7)$.

The initial pre-period consists of branches $\backslash mathcal\{B\}(5)$ and $\backslash mathcal\{B\}(6)$ with exceptional structure.

Branches $\backslash mathcal\{B\}(7)$ and $\backslash mathcal\{B\}(8)$ form the primitive period such that $\backslash mathcal\{B\}(j)\backslash simeq\backslash mathcal\{B\}(7)$, for odd $j\backslash ge\; 9$, and $\backslash mathcal\{B\}(j)\backslash simeq\backslash mathcal\{B\}(8)$, for even $j\backslash ge\; 10$.

The root of the tree is the metabelian $3$-group with identifier $R=\backslash langle\; 243,6\backslash rangle$, that is, a group of order $|\; R|=3^5=243$ and with counting number $6$. This root is not coclass settled, whence its entire descendant tree $\backslash mathcal\{T\}(R)$ is of considerably higher complexity than the coclass-$2$ subtree $\backslash mathcal\{T\}^2(R)$, whose first six branches are drawn in the diagram of Figure 4.

The additional structure can be viewed as a sort of coordinate system in which the tree is embedded. The horizontal abscissa is labelled with the transfer kernel type (TKT) $\backslash varkappa$, and the vertical ordinate is labelled with a single component $\backslash tau(1)$ of the transfer target type (TTT). The vertices of the tree are drawn in such a manner that members of periodic infinite sequences form a vertical column sharing a common TKT. On the other hand, metabelian groups of a fixed order, represented by vertices of depth at most $1$, form a horizontal row sharing a common first component of the TTT. (To discourage any incorrect interpretations, we explicitly point out that the first component of the TTT of non-metabelian groups or metabelian groups, represented by vertices of depth $2$, is usually smaller than expected, due to stabilization phenomena!) The TTT of all groups in this tree represented by a big full disk, which indicates a bicyclic centre of type $(3,3)$, is given by $\backslash tau=[\; A(3,c),(3,3,3),(9,3),(9,3)]$ with varying first component $\backslash tau(1)=A(3,c)$, the nearly homocyclic abelian $3$-group of order $3^c$, and fixed further components $\backslash tau(2)=(3,3,3)\backslash hat\{=\}(1^3)$ and $\backslash tau(3)=\backslash tau(4)=(9,3)\backslash hat\{=\}(21)$, where the abelian type invariants are either written as orders of cyclic components or as their $3$-logarithms with exponents indicating iteration. (The latter notation is employed in Figure 4.) Since the coclass of all groups in this tree is $2$, the connection between the order $3^n$ and the nilpotency class is given by $c=n-2$.

For searching a particular group in a descendant tree by looking for patterns defined by the kernels and targets of Artin transfers, it is frequently adequate to reduce the number of vertices in the branches of a dense tree with high complexity by sifting groups with desired special properties, for example

:* filtering the $\backslash sigma$-groups,

:* eliminating a set of certain transfer kernel types,

:* cancelling all non-metabelian groups (indicated by small contour squares in Fig. 4),

:* removing metabelian groups with cyclic centre (denoted by small full disks in Fig. 4),

:* cutting off vertices whose distance from the mainline (depth) exceeds some lower bound,

:* combining several different sifting criteria.

The result of such a sieving procedure is called a pruned descendant tree with respect to the desired set of properties.

However, in any case, it should be avoided that the main line of a coclass tree is eliminated, since the result would be a disconnected infinite set of finite graphs instead of a tree.

For example, it is neither recommended to eliminate all $\backslash sigma$-groups in Figure 4 nor to eliminate all groups with TKT $\backslash varkappa=(0122)$.

In Figure 4, the big double contour rectangle surrounds the pruned coclass tree $\backslash mathcal\{T\}^2\_\{\backslash ast\}(R)$, where the numerous vertices with TKT $\backslash varkappa=(2122)$ are completely eliminated. This would, for instance, be useful for searching a $\backslash sigma$-group with TKT $\backslash varkappa=(1122)$ and first component $\backslash tau(1)=(43)$ of the TTT. In this case, the search result would even be a unique group. We expand this idea further in the following detailed discussion of an important example.

The oldest example of searching for a finite p-group by the strategy of pattern recognition via Artin transfers goes back to 1934, when A. Scholz and O. Taussky

{{cite journal|

author=Scholz, A. |author2=Taussky, O.|

year=1934|

title=Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm|

journal=J. Reine Angew. Math.|

volume=171|

pages=19–41}}

tried to determine the Galois group $G=\backslash mathrm\{G\}\_3^\{\backslash infty\}(K)=\backslash mathrm\{Gal\}(\backslash mathrm\{F\}\_3^\{\backslash infty\}(K)|\; K)$ of the Hilbert $3$-class field tower, that is the maximal unramified pro-$3$ extension $\backslash mathrm\{F\}\_3^\{\backslash infty\}(K)$, of the complex quadratic number field $K=\backslash Q(\backslash sqrt\{-9748\}).$ They actually succeeded in finding the maximal metabelian quotient $Q=G/G\text{'}\text{'}=\backslash mathrm\{G\}\_3^2(K)=\backslash mathrm\{Gal\}(\backslash mathrm\{F\}\_3^2(K)|\; K)$ of $G$, that is the Galois group of the second Hilbert $3$-class field $\backslash mathrm\{F\}\_3^2(K)$ of $K$.

However, it needed $78$ years until M. R. Bush and D. C. Mayer, in 2012, provided the first rigorous proof

that the (potentially infinite) $3$-tower group $G=\backslash mathrm\{G\}\_3^\{\backslash infty\}(K)$ coincides with the finite $3$-group $\backslash mathrm\{G\}\_3^3(K)=\backslash mathrm\{Gal\}(\backslash mathrm\{F\}\_3^3(K)|\; K)$ of derived length $\backslash mathrm\{dl\}(G)=3$, and thus the $3$-tower of $K$ has exactly three stages, stopping at the third Hilbert $3$-class field $\backslash mathrm\{F\}\_3^3(K)$ of $K$.

The search is performed with the aid of the p-group generation algorithm by M. F. Newman

{{cite book|

author=Newman, M. F.|

year=1977|

title=Determination of groups of prime-power order|

publisher=pp. 73-84, in: Group Theory, Canberra, 1975, Lecture Notes in Math., Vol. 573, Springer, Berlin}}

and E. A. O'Brien.

{{cite journal|

author=O'Brien, E. A.|

year=1990|

title=The p-group generation algorithm|

journal=J. Symbolic Comput.|

volume=9|

issue=5–6|

pages=677–698

|doi=10.1016/s0747-7171(08)80082-x|

doi-access=free}}

For the initialization of the algorithm, two basic invariants must be determined. Firstly, the generator rank $d$ of the p-groups to be constructed. Here, we have $p=3$ and $d=r\_3(K)=d(\backslash mathrm\{Cl\}\_3(K))$ is given by the $3$-class rank of the quadratic field $K$. Secondly, the abelian type invariants of the $3$-class group $\backslash mathrm\{Cl\}\_3(K)\backslash simeq\; (1^2)$ of $K$. These two invariants indicate the root of the descendant tree which will be constructed successively. Although the p-group generation algorithm is designed to use the parent-descendant definition by means of the lower exponent-p central series, it can be fitted to the definition with the aid of the usual lower central series. In the case of an elementary abelian p-group as root, the difference is not very big. So we have to start with the elementary abelian $3$-group of rank two, which has the SmallGroups identifier $\backslash langle\; 9,2\backslash rangle$, and to construct the descendant tree $\backslash mathcal\{T\}(\backslash langle\; 9,2\backslash rangle)$. We do that by iterating the p-group generation algorithm, taking suitable capable descendants of the previous root as the next root, always executing an increment of the nilpotency class by a unit.

As explained at the beginning of the section Pattern recognition, we must prune the descendant tree with respect to the invariants TKT and TTT of the $3$-tower group $G$, which are determined by the arithmetic of the field $K$ as $\backslash varkappa\backslash in\backslash \{\; (2334),(2434)\backslash \}$ (exactly two fixed points and no transposition) and $\backslash tau=[\; (21)(32)(21)(21)]$. Further, any quotient of $G$ must be a $\backslash sigma$-group, enforced by number theoretic requirements for the quadratic field $K$.

The root $\backslash langle\; 9,2\backslash rangle$ has only a single capable descendant $\backslash langle\; 27,3\backslash rangle$ of type $(1^2)$. In terms of the nilpotency class, $\backslash langle\; 9,2\backslash rangle$ is the class-$1$ quotient $G/\backslash gamma\_2(G)$ of $G$ and $\backslash langle\; 27,3\backslash rangle$ is the class-$2$ quotient $G/\backslash gamma\_3(G)$ of $G$. Since the latter has nuclear rank two, there occurs a bifurcation $\backslash mathcal\{T\}(\backslash langle\; 27,3\backslash rangle)=\backslash mathcal\{T\}^1(\backslash langle\; 27,3\backslash rangle)\backslash sqcup\backslash mathcal\{T\}^2(\backslash langle\; 27,3\backslash rangle)$, where the former component $\backslash mathcal\{T\}^1(\backslash langle\; 27,3\backslash rangle)$ can be eliminated by the stabilization criterion $\backslash varkappa=(\backslash ast\; 000)$ for the TKT of all $3$-groups of maximal class.

Due to the inheritance property of TKTs, only the single capable descendant $\backslash langle\; 243,8\backslash rangle$ qualifies as the class-$3$ quotient $G/\backslash gamma\_4(G)$ of $G$.

There is only a single capable $\backslash sigma$-group $\backslash langle\; 729,54\backslash rangle$ among the descendants of $\backslash langle\; 243,8\backslash rangle$. It is the class-$4$ quotient $G/\backslash gamma\_5(G)$ of $G$ and has nuclear rank two.

This causes the essential bifurcation $\backslash mathcal\{T\}(\backslash langle\; 729,54\backslash rangle)=\backslash mathcal\{T\}^2(\backslash langle\; 729,54\backslash rangle)\backslash sqcup\backslash mathcal\{T\}^3(\backslash langle\; 729,54\backslash rangle)$ in two subtrees belonging to different coclass graphs $\backslash mathcal\{G\}(3,2)$ and $\backslash mathcal\{G\}(3,3)$. The former contains the metabelian quotient $Q=G/G\text{'}\text{'}$ of $G$ with two possibilities $Q\backslash in\backslash \{\backslash langle\; 2187,302\backslash rangle,\backslash langle\; 2187,306\backslash rangle\backslash \}$, which are not balanced with relation rank $r=3>2=d$ bigger than the generator rank. The latter consists entirely of non-metabelian groups and yields the desired $3$-tower group $G$ as one among the two Schur $\backslash sigma$-groups $\backslash langle\; 729,54\backslash rangle-\backslash \#2;2$ and $\backslash langle\; 729,54\backslash rangle-\backslash \#2;6$ with $r=2=d$.

Finally the termination criterion is reached at the capable vertices $\backslash langle\; 2187,303\backslash rangle-\backslash \#1;1\backslash in\backslash mathcal\{G\}(3,2)$ and $\backslash langle\; 729,54\backslash rangle-\backslash \#2;3-\backslash \#1;1\backslash in\backslash mathcal\{G\}(3,3)$, since the TTT $\backslash tau=[\; (21)(3^2)(21)(21)]>[\; (21)(32)(21)(21)]$ is too big and will even increase further, never returning to $[\; (21)(32)(21)(21)]$. The complete search process is visualized in Table 1, where, for each of the possible successive p-quotients $P\_c=G/\backslash gamma\_\{c+1\}(G)$ of the $3$-tower group $G=\backslash mathrm\{G\}\_3^\{\backslash infty\}(K)$ of $K=\backslash Q(\backslash sqrt\{-9748\})$, the nilpotency class is denoted by $c=\backslash mathrm\{cl\}(P\_c)$, the nuclear rank by $\backslash nu=\backslash nu(P\_c)$, and the p-multiplicator rank by $\backslash mu=\backslash mu(P\_c)$.

This section shows exemplarily how commutator calculus can be used for determining the kernels and targets of Artin transfers explicitly. As a concrete example we take the metabelian $3$-groups with bicyclic centre, which are represented by big full disks as vertices, of the coclass tree diagram in Figure 4. They form ten periodic infinite sequences, four, resp. six, for even, resp. odd, nilpotency class $c$, and can be characterized with the aid of a parametrized polycyclic power-commutator presentation:

{{EquationNote|1}}

& x^3=s_c^w,\ y^3=s_3^2s_4s_c^z,\ s_j^3=s_{j+2}^2s_{j+3}\text{ for }2\le j\le c-3,\ s_{c-2}^3=s_c^2,\ t_3^3=1,\\

& s_2=[ y,x],\ t_3=[ s_2,y],\ s_j=[ s_{j-1},x]\text{ for }3\le j\le c\rangle,\end{align}

where $c\backslash ge\; 5$ is the nilpotency class, $3^n$ with $n=c+2$ is the order, and $0\backslash le\; w\backslash le\; 1,\; -1\backslash le\; z\backslash le\; 1$ are parameters.

The transfer target type (TTT) of the group $G=G^\{c,n\}(z,w)$ depends only on the nilpotency class $c$, is independent of the parameters $w,z$, and is given uniformly by $\backslash tau=[\; A(3,c),(3,3,3),(9,3),(9,3)]$. This phenomenon is called a polarization, more precisely a uni-polarization, at the first component.

The transfer kernel type (TKT) of the group $G=G^\{c,n\}(z,w)$ is independent of the nilpotency class $c$, but depends on the parameters $w,z$, and is given by c.18, $\backslash varkappa=(0122)$, for $w=z=0$ (a mainline group), H.4, $\backslash varkappa=(2122)$, for $w=0,z=\backslash pm\; 1$ (two capable groups), E.6, $\backslash varkappa=(1122)$, for $w=1,z=0$ (a terminal group), and E.14, $\backslash varkappa\backslash in\backslash \{\; (4122),(3122)\backslash \}$, for $w=1,z=\backslash pm\; 1$ (two terminal groups). For even nilpotency class, the two groups of types H.4 and E.14, which differ in the sign of the parameter $z$ only, are isomorphic.

These statements can be deduced by means of the following considerations.

As a preparation, it is useful to compile a list of some commutator relations, starting with those given in the presentation,

$[\; a,x]=1$ for $a\backslash in\backslash \{\; s\_c,t\_3\backslash \}$ and $[\; a,y]=1$ for $a\backslash in\backslash \{\; s\_3,\backslash ldots,s\_c,t\_3\backslash \}$,

which shows that the bicyclic centre is given by $\backslash zeta\_1(G)=\backslash langle\; s\_c,t\_3\backslash rangle$. By means of the right product rule $[\; a,xy]=[\; a,y]\backslash cdot[\; a,x]\backslash cdot[[a,x],y]$ and the right power rule $[\; a,y^2]=[\; a,y]^\{1+y\}$,

we obtain $[\; s\_2,xy]=s\_3t\_3$, $[\; s\_2,xy^2]=s\_3t\_3^2$, and $[\; s\_j,xy]=[\; s\_j,xy^2]=[\; s\_j,x]=s\_\{j+1\}$, for $j\backslash ge\; 3$.

The maximal subgroups of $G$ are taken in a similar way as in the section on the computational implementation, namely

:$\backslash begin\{align\}$

H_1 &=\langle y,G'\rangle \\

H_2 &=\langle x,G'\rangle \\

H_3 &=\langle xy,G'\rangle \\

H_4 &=\langle xy^2,G'\rangle \\

\end{align}

Their derived subgroups are crucial for the behavior of the Artin transfers. By making use of the general formula $H\_i\text{'}=(G\text{'})^\{h\_i-1\}$, where $H\_i=\backslash langle\; h\_i,G\text{'}\backslash rangle$, and where we know that $G\text{'}=\backslash langle\; s\_2,\; t\_3,\; s\_3,\; \backslash ldots,\; s\_c\backslash rangle$ in the present situation, it follows that

:$\backslash begin\{align\}$

H_1' &=\left \langle s_2^{y-1} \right \rangle= \left \langle t_3 \right \rangle \\

H_2' &=\left \langle s_2^{x-1},\ldots,s_{c-1}^{x-1}\right \rangle=\left \langle s_3,\ldots,s_c\right \rangle \\

H_3' &=\left \langle s_2^{xy-1},\ldots,s_{c-1}^{xy-1}\right \rangle=\left \langle s_3t_3,s_4,\ldots,s_c\right \rangle \\

H_4' &=\left \langle s_2^{xy^2-1},\ldots,s_{c-1}^{xy^2-1}\right \rangle=\left \langle s_3t_3^2,s_4,\ldots,s_c \right \rangle

\end{align}

Note that $H\_1$ is not far from being abelian, since $H\_1\text{'}=\backslash langle\; t\_3\backslash rangle$ is contained in the centre $\backslash zeta\_1(G)=\backslash langle\; s\_c,t\_3\backslash rangle$.

As the first main result, we are now in the position to determine the abelian type invariants of the derived quotients:

:$H\_1/H\_1\text{'}=\backslash langle\; y,s\_2,\backslash ldots,s\_c\backslash rangle\; H\_1\text{'}/H\_1\text{'}\backslash simeq\; A(3,c),$

the unique quotient which grows with increasing nilpotency class $c$, since $\backslash mathrm\{ord\}(y)=\backslash mathrm\{ord\}(s\_2)\; =3^m$ for even $c=2m$ and $\backslash mathrm\{ord\}(y)=3^\{m+1\},\backslash mathrm\{ord\}(s\_2)=3^m$ for odd $c=2m+1$,

:$\backslash begin\{align\}$

H_2/H_2' &=\langle x,s_2,t_3\rangle H_2'/H_2'\simeq (3,3,3) \\

H_3/H_3' &=\langle xy,s_2,t_3\rangle H_3'/H_3'\simeq (9,3) \\

H_4/H_4' &=\langle xy^2,s_2,t_3\rangle H_4'/H_4'\simeq (9,3)

\end{align}

since generally $\backslash mathrm\{ord\}(s\_2)=\backslash mathrm\{ord\}(t\_3)=3$, but $\backslash mathrm\{ord\}(x)=3$ for $H\_2$, whereas $\backslash mathrm\{ord\}(xy)=\backslash mathrm\{ord\}(xy^2)=9$ for $H\_3$ and $H\_4$.

Now we come to the kernels of the Artin transfer homomorphisms $T\_i:\; G\backslash to\; H\_i/H\_i\text{'}$. It suffices to investigate the induced transfers $\backslash tilde\{T\}\_i:\; G/G\text{'}\backslash to\; H\_i/H\_i\text{'}$ and to begin by finding expressions for the images $\backslash tilde\{T\}\_i(gG\text{'})$ of elements $gG\text{'}\backslash in\; G/G\text{'}$, which can be expressed in the form

:$g\backslash equiv\; x^j\; y^\{\backslash ell\}\; \backslash pmod\{G\text{'}\},\; \backslash qquad\; j,\; \backslash ell\; \backslash in\; \backslash \{-1,\; 0,\; 1\backslash \}.$

First, we exploit outer transfers as much as possible:

:$\backslash begin\{align\}$

x \notin H_1 &\Rightarrow \tilde{T}_1(xG')=x^3H_1'=s_c^wH_1' \\

y \notin H_2 &\Rightarrow \tilde{T}_2(yG')=y^3H_2'=s_3^2s_4s_c^zH_2'=1\cdot H_2' \\

x,y \notin H_3,H_4 &\Rightarrow \begin{cases} \tilde{T}_i(xG')=x^3H_i'=s_c^wH_i'=1\cdot H_i' \\ \tilde{T}_i(yG') =y^3H_i'=s_3^2s_4s_c^zH_i'=s_3^2H_i'\end{cases} && 3\le i\le 4

\end{align}

Next, we treat the unavoidable inner transfers, which are more intricate. For this purpose, we use the polynomial identity

:$X^2+X+1=(X-1)^2+3(X-1)+3$

to obtain:

:$\backslash begin\{align\}$

y\in H_1 &\Rightarrow \tilde{T}_1(yG')=y^{1+x+x^2}H_1'=y^{3+3(x-1)+(x-1)^2}H_1'=y^3\cdot[ y,x]^3\cdot[[ y,x],x] H_1' = s_3^2s_4s_c^zs_2^3s_3H_1'=s_2^3s_3^3s_4s_c^zH_1'=s_c^zH_1' \\

x\in H_2 &\Rightarrow \tilde{T}_2(xG')=x^{1+y+y^2}H_2'=x^{3+3(y-1)+(y-1)^2}H_2'=x^3\cdot[ x,y]^3\cdot[[ x,y],y] H_2'=s_c^ws_2^{-3}t_3^{-1}H_2'=t_3^{-1}H_2'

\end{align}

Finally, we combine the results: generally

:$\backslash tilde\{T\}\_i(gG\text{'})=\backslash tilde\{T\}\_i(xG\text{'})^j\backslash tilde\{T\}\_i(yG\text{'})^\{\backslash ell\},$

and in particular,

:$\backslash begin\{align\}$

\tilde{T}_1(gG') &=s_c^{wj+z\ell}H_1' \\

\tilde{T}_2(gG') &=t_3^{-j}H_2' \\

\tilde{T}_i(gG') &=s_3^{2\ell}H_i' && 3\le i\le 4

\end{align}

To determine the kernels, it remains to solve the equations:

:$\backslash begin\{align\}$

s_c^{wj+z\ell}H_1'=H_1' &\Rightarrow \begin{cases} \text{arbitrary } j, \ell \text{ and } w=z=0 \\ \ell=0, \text{arbitrary } j \text{ and } w=0,z=\pm 1 \\ j=0, \text{arbitrary } \ell \text{ and } w=1,z=0 \\ j=\mp \ell, w=1,z=\pm 1 \end{cases} \\

t_3^{-j}H_2'=H_2' &\Rightarrow j=0 \text{ with arbitrary } \ell \\

s_3^{2\ell}H_i'=H_i' &\Rightarrow \ell=0 \text{ with arbitrary } j && 3\le i\le 4

\end{align}

The following equivalences, for any $1\backslash le\; i\backslash le\; 4$, finish the justification of the statements:

• $j,\backslash ell$ both arbitrary $\backslash Leftrightarrow\; \backslash ker(T\_i)=\backslash langle\; x,y,G\text{'}\backslash rangle=G\; \backslash Leftrightarrow\; \backslash varkappa(i)=0$.
• $j=0$ with arbitrary $\backslash ell\; \backslash Leftrightarrow\; \backslash ker(T\_i)=\backslash langle\; y,G\text{'}\backslash rangle=H\_1\; \backslash Leftrightarrow\; \backslash varkappa(i)=1$,
• $\backslash ell=0$ with arbitrary $j\; \backslash Leftrightarrow\; \backslash ker(T\_i)=\backslash langle\; x,G\text{'}\backslash rangle=H\_2\; \backslash Leftrightarrow\; \backslash varkappa(i)=2$,
• $j=\backslash ell\; \backslash Leftrightarrow\; \backslash ker(T\_i)=\backslash langle\; xy,G\text{'}\backslash rangle=H\_3\; \backslash Leftrightarrow\; \backslash varkappa(i)=3$,
• $j=-\backslash ell\; \backslash Leftrightarrow\; \backslash ker(T\_i)=\backslash langle\; xy^\{-1\},G\text{'}\backslash rangle=H\_4\; \backslash Leftrightarrow\; \backslash varkappa(i)=4$

Consequently, the last three components of the TKT are independent of the parameters $w,z,$ which means that both, the TTT and the TKT, reveal a uni-polarization at the first component.

The aim of this section is to present a collection of structured coclass trees (SCTs) of finite p-groups with parametrized presentations and a succinct summary of invariants.

The underlying prime $p$ is restricted to small values $p\backslash in\backslash \{\; 2,3,5\backslash \}$.

The trees are arranged according to increasing coclass $r\backslash ge\; 1$ and different abelianizations within each coclass.

To keep the descendant numbers manageable, the trees are pruned by eliminating vertices of depth bigger than one.

Further, we omit trees where stabilization criteria enforce a common TKT of all vertices, since we do not consider such trees as structured any more.

The invariants listed include

:* pre-period and period length,

:* depth and width of branches,

:* uni-polarization, TTT and TKT,

:* $\backslash sigma$-groups.

We refrain from giving justifications for invariants, since the way how invariants are derived from presentations was demonstrated exemplarily in the section on commutator calculus

File:Coclass1Tree2Groups.png

For each prime $p\backslash in\backslash \{\; 2,3,5\backslash \}$, the unique tree of p-groups of maximal class is endowed with information on TTTs and TKTs, that is, $\backslash mathcal\{T\}^1(\backslash langle\; 4,2\backslash rangle)$ for $p=2,\; \backslash mathcal\{T\}^1(\backslash langle\; 9,2\backslash rangle)$ for $p=3$, and $\backslash mathcal\{T\}^1\; (\backslash langle\; 25,2\backslash rangle)$ for $p=5$. In the last case, the tree is restricted to metabelian $5$-groups.

The $2$-groups of coclass $1$ in Figure 5 can be defined by the following parametrized polycyclic pc-presentation, quite different from Blackburn's presentation.

{{EquationNote|2}}

& x^2=s_c^w,\ y^2=s_c^z,\ s_j^2=s_{j+1}s_{j+2}\text{ for }2\le j\le c-2,\ s_{c-1}^2=s_c,\\

& s_2=[ y,x],\ s_j=[ s_{j-1},x]=[ s_{j-1},y]\text{ for }3\le j\le c\rangle,\end{align}

where the nilpotency class is $c\backslash ge\; 3$, the order is $2^n$ with $n=c+1$, and $w,z$ are parameters. The branches are strictly periodic with pre-period $1$ and period length $1$, and have depth $1$ and width $3$.

Polarization occurs for the third component and the TTT is $\backslash tau=[(1^2),(1^2),A(2,c)]$, only dependent on $c$ and with cyclic $A(2,c)$. The TKT depends on the parameters and is $\backslash varkappa=(210)$ for the dihedral mainline vertices with $w=z=0$,

$\backslash varkappa=(213)$ for the terminal generalized quaternion groups with $w=z=1$, and $\backslash varkappa=(211)$ for the terminal semi dihedral groups with $w=1,z=0$. There are two exceptions, the abelian root with $\backslash tau=[\; (1),(1),(1)]$ and $\backslash varkappa=(000)$, and the usual quaternion group with $\backslash tau=[\; (2),(2),(2)]$ and $\backslash varkappa=(123)$.

File:Coclass1Tree3Groups.png

The $3$-groups of coclass $1$ in Figure 6 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Blackburn's presentation.

{{EquationNote|3}}

& x^3=s_c^w,\ y^3=s_3^2s_4s_c^z,\ t_3=s_c^a,\ s_j^3=s_{j+2}^2s_{j+3}\text{ for }2\le j\le c-3,\ s_{c-2}^3=s_c^2,\\

& s_2=[ y,x],\ t_3=[ s_2,y],\ s_j=[ s_{j-1},x]\text{ for }3\le j\le c\rangle,\end{align}

where the nilpotency class is $c\backslash ge\; 5$, the order is $3^n$ with $n=c+1$, and $a,w,z$ are parameters. The branches are strictly periodic with pre-period $2$ and period length $2$, and have depth $1$ and width $7$. Polarization occurs for the first component and the TTT is $\backslash tau=[\; A(3,c-a),(1^2),(1^2),(1^2)]$, only dependent on $c$ and $a$. The TKT depends on the parameters and is $\backslash varkappa=(0000)$ for the mainline vertices with $a=w=z=0,\; \backslash varkappa=(1000)$ for the terminal vertices with $a=0,w=1,z=0,\; \backslash varkappa=(2000)$ for the terminal vertices with $a=w=0,z=\backslash pm\; 1$, and $\backslash varkappa=(0000)$ for the terminal vertices with $a=1,w\backslash in\backslash \{\; -1,0,1\backslash \},z=0$. There exist three exceptions, the abelian root with $\backslash tau=[\; (1),(1),(1),(1)]$, the extra special group of exponent $9$ with $\backslash tau=[\; (1^2),(2),(2),(2)]$ and $\backslash varkappa=(1111)$, and the Sylow $3$-subgroup of the alternating group $A\_9$ with $\backslash tau=[\; (1^3),(1^2),(1^2),(1^2)]$. Mainline vertices and vertices on odd branches are $\backslash sigma$-groups.

File:Coclass1Tree5Groups.png

The metabelian $5$-groups of coclass $1$ in Figure 7 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Miech's presentation.{{cite journal|author=Miech, R. J.|year=1970|title=Metabelian p-groups of maximal class|journal=Trans. Amer. Math. Soc.|volume=152|issue=2|pages=331–373|doi=10.1090/s0002-9947-1970-0276343-7|doi-access=free}}

{{EquationNote|4}}

& x^5=s_c^w,\ y^5=s_c^z,\ t_3=s_c^a,\\

& s_2=[ y,x],\ t_3=[ s_2,y],\ s_j=[ s_{j-1},x]\text{ for }3\le j\le c\rangle,\end{align}

where the nilpotency class is $c\backslash ge\; 3$, the order is $5^n$ with $n=c+1$, and $a,w,z$ are parameters. The (metabelian!) branches are strictly periodic with pre-period $3$ and period length $4$, and have depth $3$ and width $67$. (The branches of the complete tree, including non-metabelian groups, are only virtually periodic and have bounded width but unbounded depth!) Polarization occurs for the first component and the TTT is $\backslash tau=[\; A(5,c-k),(1^2)^5]$, only dependent on $c$ and the defect of commutativity $k$. The TKT depends on the parameters and is $\backslash varkappa=(0^6)$ for the mainline vertices with $a=w=z=0,\; \backslash varkappa=(10^5)$ for the terminal vertices with $a=0,w=1,z=0,\; \backslash varkappa=(20^5)$ for the terminal vertices with $a=w=0,\; z\backslash ne\; 0$, and $\backslash varkappa=(0^6)$ for the vertices with $a\backslash ne\; 0$. There exist three exceptions, the abelian root with $\backslash tau=[(1)^6]$, the extra special group of exponent $25$ with $\backslash tau=[(1^2),(2)^5]$ and $\backslash varkappa=(1^6)$, and the group $\backslash langle\; 15625,631\backslash rangle$ with $\backslash tau=[\; (1^5),(1^2)^5]$. Mainline vertices and vertices on odd branches are $\backslash sigma$-groups.

Three coclass trees, $\backslash mathcal\{T\}^2(\backslash langle\; 243,6\backslash rangle)$, $\backslash mathcal\{T\}^2(\backslash langle\; 243,8\backslash rangle)$ and $\backslash mathcal\{T\}^2(\backslash langle\; 729,40\backslash rangle)$ for $p=3$, are endowed with information concerning TTTs and TKTs.

File:Coclass2TreeQType33.png

On the tree $\backslash mathcal\{T\}^2(\backslash langle\; 243,6\backslash rangle)$, the $3$-groups of coclass $2$ with bicyclic centre in Figure 8 can be defined by the following parametrized polycyclic pc-presentation.

{{EquationNote|5}}

& x^3=s_c^w,\ y^3=s_3^2s_4s_c^z,\ s_j^3=s_{j+2}^2s_{j+3}\text{ for }2\le j\le c-3,\ s_{c-2}^3=s_c^2,\ t_3^3=1,\\

& s_2=[ y,x],\ t_3=[ s_2,y],\ s_j=[ s_{j-1},x]\text{ for }3\le j\le c\rangle,\end{align}

where the nilpotency class is $c\backslash ge\; 5$, the order is $3^n$ with $n=c+2$, and $w,z$ are parameters.

The branches are strictly periodic with pre-period $2$ and period length $2$, and have depth $3$ and width $18$.

Polarization occurs for the first component and the TTT is $\backslash tau=[\; A(3,c),(1^3),(21),(21)]$, only dependent on $c$.

The TKT depends on the parameters and is

$\backslash varkappa=(0122)$ for the mainline vertices with $w=z=0$,

$\backslash varkappa=(2122)$ for the capable vertices with $w=0,z=\backslash pm\; 1$,

$\backslash varkappa=(1122)$ for the terminal vertices with $w=1,z=0$,

and $\backslash varkappa=(3122)$ for the terminal vertices with $w=1,z=\backslash pm\; 1$.

Mainline vertices and vertices on even branches are $\backslash sigma$-groups.

File:Coclass2TreeUType33.png

On the tree $\backslash mathcal\{T\}^2(\backslash langle\; 243,8\backslash rangle)$, the $3$-groups of coclass $2$ with bicyclic centre in Figure 9 can be defined by the following parametrized polycyclic pc-presentation.

{{EquationNote|6}}

& y^3=s_3t_c^w,\ x^3=t_3t_4^2t_5t_c^z,\ t_j^3=t_{j+2}^2t_{j+3}\text{ for }2\le j\le c-3,\ t_{c-2}^3=t_c^2,\ s_3^3=1,\\

& t_2=[ y,x],\ s_3=[ t_2,x],\ t_j=[ t_{j-1},y]\text{ for }3\le j\le c\rangle,\end{align}

where the nilpotency class is $c\backslash ge\; 6$, the order is $3^n$ with $n=c+2$, and $w,z$ are parameters.

The branches are strictly periodic with pre-period $2$ and period length $2$, and have depth $3$ and width $16$.

Polarization occurs for the second component and the TTT is $\backslash tau=[\; (21),A(3,c),(21),(21)]$, only dependent on $c$.

The TKT depends on the parameters and is

$\backslash varkappa=(2034)$ for the mainline vertices with $w=z=0$,

$\backslash varkappa=(2134)$ for the capable vertices with $w=0,z=\backslash pm\; 1$,

$\backslash varkappa=(2234)$ for the terminal vertices with $w=1,z=0$,

and $\backslash varkappa=(2334)$ for the terminal vertices with $w=1,z=\backslash pm\; 1$.

Mainline vertices and vertices on even branches are $\backslash sigma$-groups.

$\backslash mathcal\{T\}^2(\backslash langle\; 16,3\backslash rangle)$ and $\backslash mathcal\{T\}^2(\backslash langle\; 16,4\backslash rangle)$ for $p=2$,

$\backslash mathcal\{T\}^2(\backslash langle\; 243,15\backslash rangle)$ and $\backslash mathcal\{T\}^2(\backslash langle\; 243,17\backslash rangle)$ for $p=3$.

$\backslash mathcal\{T\}^2(\backslash langle\; 16,11\backslash rangle)$ for $p=2$, and $\backslash mathcal\{T\}^2(\backslash langle\; 81,12\backslash rangle)$ for $p=3$.

$\backslash mathcal\{T\}^3(\backslash langle\; 729,13\backslash rangle)$, $\backslash mathcal\{T\}^3(\backslash langle\; 729,18\backslash rangle)$ and $\backslash mathcal\{T\}^3(\backslash langle\; 729,21\backslash rangle)$ for $p=3$.

$\backslash mathcal\{T\}^3(\backslash langle\; 32,35\backslash rangle)$ and $\backslash mathcal\{T\}^3(\backslash langle\; 64,181\backslash rangle)$ for $p=2$,

$\backslash mathcal\{T\}^3(\backslash langle\; 243,38\backslash rangle)$ and $\backslash mathcal\{T\}^3(\backslash langle\; 243,41\backslash rangle)$ for $p=3$.

File:MinDiscriminantsTreeQ.png

In algebraic number theory and class field theory, structured descendant trees (SDTs) of finite p-groups provide an excellent tool for

• visualizing the location of various non-abelian p-groups $G(K)$ associated with algebraic number fields $K$,
• displaying additional information about the groups $G(K)$ in labels attached to corresponding vertices, and
• emphasizing the periodicity of occurrences of the groups $G(K)$ on branches of coclass trees.

For instance, let $p$ be a prime number, and assume that $F^2\_p(K)$ denotes the second Hilbert p-class field of an algebraic number field $K$, that is the maximal metabelian unramified extension of $K$ of degree a power of $p$. Then the second p-class group $G^2\_p(K)=\backslash mathrm\{Gal\}(F^2\_p(K)|\; K)$ of $K$ is usually a non-abelian p-group of derived length $2$ and frequently permits to draw conclusions about the entire p-class field tower of $K$, that is the Galois group $G^\{\backslash infty\}\_p(K)=\backslash mathrm\{Gal\}(F^\{\backslash infty\}\_p(K)|\; K)$ of the maximal unramified pro-p extension $F^\{\backslash infty\}\_p(K)$ of $K$.

Given a sequence of algebraic number fields $K$ with fixed signature $(r\_1,r\_2)$, ordered by the absolute values of their discriminants $d=d(K)$, a suitable structured coclass tree (SCT) $\backslash mathcal\{T\}$, or also the finite sporadic part $\backslash mathcal\{G\}\_0(p,r)$ of a coclass graph $\backslash mathcal\{G\}(p,r)$, whose vertices are entirely or partially realized by second p-class groups $G^2\_p(K)$ of the fields $K$ is endowed with additional arithmetical structure when each realized vertex $V\backslash in\backslash mathcal\{T\}$, resp. $V\backslash in\backslash mathcal\{G\}\_0(p,r)$, is mapped to data concerning the fields $K$ such that $V=G^2\_p(K)$.

File:MinDiscriminantsTreeU.png

To be specific, let $p=3$ and consider complex quadratic fields $K(d)=\backslash Q(\backslash sqrt\{d\})$ with fixed signature $(0,1)$ having $3$-class groups with type invariants $(3,3)$. See OEIS A242863 [http://oeis.org/A242863]. Their second $3$-class groups $G^2\_3(K)$ have been determined by D. C. Mayer

for the range

{{cite journal| author=Boston, N. |author2=Bush, M. R. |author3=Hajir, F.| year=2015| title=Heuristics for p-class towers of imaginary quadratic fields| journal=Math. Ann.| arxiv=1111.4679}} for the extended range

Let us firstly select the two structured coclass trees (SCTs) $\backslash mathcal\{T\}^2(\backslash langle\; 243,6\backslash rangle)$ and $\backslash mathcal\{T\}^2\; (\backslash langle\; 243,8\backslash rangle)$, which are known from Figures 8 and 9 already, and endow these trees with additional arithmetical structure by surrounding a realized vertex $V$ with a circle and attaching an adjacent underlined boldface integer $\backslash min\backslash$

Concerning the periodicity of occurrences of second $3$-class groups $G^2\_3(K(d))$ of complex quadratic fields, it was proved that only every other branch of the trees in Figures 10 and 11 can be populated by these metabelian $3$-groups and that the distribution sets in with a ground state (GS) on branch $\backslash mathcal\{B\}(6)$ and continues with higher excited states (ES) on the branches $\backslash mathcal\{B\}(j)$ with even $j\backslash ge\; 8$. This periodicity phenomenon is underpinned by three sequences with fixed TKTs

• E.14 $\backslash varkappa=(3122)$, OEIS A247693 [http://oeis.org/A247693],
• E.6 $\backslash varkappa=(1122)$, OEIS A247692 [http://oeis.org/A247692],
• H.4 $\backslash varkappa=(2122)$, OEIS A247694 [http://oeis.org/A247694]

on the ASCT $\backslash mathcal\{T\}^2(\backslash langle\; 243,6\backslash rangle)$, and by three sequences with fixed TKTs

• E.9 $\backslash varkappa=(2334)$, OEIS A247696 [http://oeis.org/A247696],
• E.8 $\backslash varkappa=(2234)$, OEIS A247695 [http://oeis.org/A247695],
• G.16 $\backslash varkappa=(2134)$,OEIS A247697 [http://oeis.org/A247697]

on the ASCT $\backslash mathcal\{T\}^2(\backslash langle\; 243,8\backslash rangle)$. Up to now, the ground state and three excited states are known for each of the six sequences, and for TKT E.9 $\backslash varkappa=(2334)$ even the fourth excited state occurred already. The minimal absolute discriminants of the various states of each of the six periodic sequences are presented in Table 2. Data for the ground states (GS) and the first excited states (ES1) has been taken from D. C. Mayer, most recent information on the second, third and fourth excited states (ES2, ES3, ES4) is due to N. Boston, M. R. Bush and F. Hajir.

File:FrequencyCoclass2Type33Sporadic.tiff

In contrast, let us secondly select the sporadic part $\backslash mathcal\{G\}\_0(3,2)$ of the coclass graph $\backslash mathcal\{G\}(3,2)$ for demonstrating that another way of attaching additional arithmetical structure to descendant trees is to display the counter below a given upper bound $b$, for instance $b=10^8$. With respect to the total counter $276\; 375$ of all complex quadratic fields with $3$-class group of type $(3,3)$ and discriminant , this gives the relative frequency as an approximation to the asymptotic density of the population in Figure 12 and Table 3. Exactly four vertices of the finite sporadic part $\backslash mathcal\{G\}\_0(3,2)$ of $\backslash mathcal\{G\}(3,2)$ are populated by second $3$-class groups $G^2\_3(K(d))$:

• $\backslash langle\; 243,5\backslash rangle$, OEIS A247689 [http://oeis.org/A247689],
• $\backslash langle\; 243,7\backslash rangle$, OEIS A247690 [http://oeis.org/A247690],
• $\backslash langle\; 729,45\backslash rangle$, OEIS A242873 [http://oeis.org/A242873],
• $\backslash langle\; 729,57\backslash rangle$, OEIS A247688 [http://oeis.org/A247688].

File:MinDiscriminantsCoclass2Type33Sporadic.tiff

File:MinDiscriminantsCoclass2Type55Sporadic.png

File:MinDiscriminantsCoclass2Type77Sporadic.tiff

Now let $p\backslash in\backslash \{\; 3,5,7\backslash \}$ and consider complex quadratic fields $K(d)=\backslash Q(\backslash sqrt\{d\})$ with fixed signature $(0,1)$ and p-class groups of type $(p,p)$. The dominant part of the second p-class groups of these fields populates the top vertices of order $p^5$ of the sporadic part $\backslash mathcal\{G\}\_0(p,2)$ of the coclass graph $\backslash mathcal\{G\}(p,2)$, which belong to the stem of P. Hall's isoclinism family $\backslash Phi\_6$, or their immediate descendants of order $p^6$. For primes $p>3$, the stem of $\backslash Phi\_6$ consists of $p+7$ regular p-groups and reveals a rather uniform behaviour with respect to TKTs and TTTs, but the seven $3$-groups in the stem of $\backslash Phi\_6$ are irregular. We emphasize that there also exist several ($3$ for $p=3$ and $4$ for $p>3$) infinitely capable vertices in the stem of $\backslash Phi\_6$ which are partially roots of coclass trees. However, here we focus on the sporadic vertices which are either isolated Schur $\backslash sigma$-groups ($2$ for $p=3$ and $p+1$ for $p>3$) or roots of finite trees within $\backslash mathcal\{G\}\_0(p,2)$ ($2$ for each $p\backslash ge\; 3$). For $p>3$, the TKT of Schur $\backslash sigma$-groups is a permutation whose cycle decomposition does not contain transpositions, whereas the TKT of roots of finite trees is a compositum of disjoint transpositions having an even number ($0$ or $2$) of fixed points.

We endow the forest $\backslash mathcal\{G\}\_0(p,2)$ (a finite union of descendant trees) with additional arithmetical structure by attaching the minimal absolute discriminant $\backslash min\backslash \{|d|\backslash mid\; V=G^2\_p(K(d))\backslash \}$ to each realized vertex $V\backslash in\backslash mathcal\{G\}\_0(p,2)$. The resulting structured sporadic coclass graph is shown in Figure 13 for $p=3$, in Figure 14 for $p=5$, and in Figure 15 for $p=7$.

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Category:Group theory

Category:Class field theory