Powerful p-group

In mathematics, in the field of group theory, especially in the study of p-groups and pro-p-groups, the concept of powerful p-groups plays an important role. They were introduced in {{harv|Lubotzky|Mann|1987}}, where a number of applications are given, including results on Schur multipliers. Powerful p-groups are used in the study of automorphisms of p-groups {{harv|Khukhro|1998}}, the solution of the restricted Burnside problem {{harv|Vaughan-Lee|1993}}, the classification of finite p-groups via the coclass conjectures {{harv|Leedham-Green|McKay|2002}}, and provided an excellent method of understanding analytic pro-p-groups {{harv|Dixon|du Sautoy|Mann|Segal|1991}}.

Formal definition

A finite p-group $G$ is called powerful if the commutator subgroup $[G,G]$ is contained in the subgroup $G^p = \langle g^p | g\in G\rangle$ for odd $p$ , or if $[G,G]$ is contained in the subgroup $G^4$ for $p=2$ .

Properties of powerful ''p''-groups

Powerful p-groups have many properties similar to abelian groups, and thus provide a good basis for studying p-groups. Every finite p-group can be expressed as a section of a powerful p-group.

Powerful p-groups are also useful in the study of pro-p groups as it provides a simple means for characterising p-adic analytic groups (groups that are manifolds over the p-adic numbers): A finitely generated pro-p group is p-adic analytic if and only if it contains an open normal subgroup that is powerful: this is a special case of a deep result of Michel Lazard (1965).

Some properties similar to abelian p-groups are: if $G$ is a powerful p-group then:

The Frattini subgroup $\Phi(G)$ of $G$ has the property $\Phi(G) = G^p.$
$G^{p^k} = \{g^{p^k}|g\in G\}$ for all $k\geq 1.$ That is, the group generated by $p$ th powers is precisely the set of $p$ th powers.
If $G = \langle g_1, \ldots, g_d\rangle$ then $G^{p^k} = \langle g_1^{p^k},\ldots,g_d^{p^k}\rangle$ for all $k\geq 1.$
The $k$ th entry of the lower central series of $G$ has the property $\gamma_k(G)\leq G^{p^{k-1}}$ for all $k\geq 1.$
Every quotient group of a powerful p-group is powerful.
The Prüfer rank of $G$ is equal to the minimal number of generators of $G.$

Some less abelian-like properties are: if $G$ is a powerful p-group then:

$G^{p^k}$ is powerful.
Subgroups of $G$ are not necessarily powerful.

References

Lazard, Michel (1965), Groupes analytiques p-adiques, Publ. Math. IHÉS 26 (1965), 389–603.
{{citation |mr=1152800 | last1=Dixon | first1=J. D. | last2=du Sautoy | first2=M. P. F. | author2-link=Marcus du Sautoy | last3=Mann | first3=A. | last4=Segal | first4=D. | title=Analytic pro-p-groups | publisher=Cambridge University Press | year=1991 | isbn=0-521-39580-1}}
{{citation |mr=1615819 | last1=Khukhro | first1=E. I. | title=p-automorphisms of finite p-groups | publisher=Cambridge University Press | year=1998 | isbn=0-521-59717-X | doi=10.1017/CBO9780511526008}}
{{Citation | last1=Leedham-Green | first1=C. R. | author1-link=Charles Leedham-Green | last2=McKay | first2=Susan | title=The structure of groups of prime power order | publisher=Oxford University Press | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853548-5 |mr=1918951 | year=2002 | volume=27}}
{{citation |mr=0873681 | last1=Lubotzky | author1-link = Alexander Lubotzky | first1=Alexander | last2=Mann | first2=Avinoam | title=Powerful p-groups. I. Finite Groups | journal=J. Algebra | volume=105 | year=1987 | pages=484–505 | doi=10.1016/0021-8693(87)90211-0 | issue=2| doi-access= }}
{{citation |mr=1364414 | last1=Vaughan-Lee | first1=Michael | title=The restricted Burnside problem | edition=2nd | publisher=Oxford University Press | year=1993 | isbn=0-19-853786-7 }}

Category:P-groups

Category:Properties of groups