inner automorphism

{{Short description|Automorphism of a group, ring, or algebra given by the conjugation action of one of its elements}}

In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.

Definition

If {{mvar|G}} is a group and {{mvar|g}} is an element of {{mvar|G}} (alternatively, if {{mvar|G}} is a ring, and {{mvar|g}} is a unit), then the function

: $\begin{align}
\varphi_g\colon G&\to G \\
\varphi_g(x)&:= g^{-1}xg
\end{align}$

is called (right) conjugation by {{mvar|g}} (see also conjugacy class). This function is an endomorphism of {{mvar|G}}: for all $x_1,x_2\in G,$

: $\varphi_g(x_1 x_2) = g^{-1} x_1 x_2g = g^{-1} x_1 \left(g g^{-1}\right) x_2 g = \left(g^{-1} x_1 g\right)\left(g^{-1} x_2 g\right) = \varphi_g(x_1)\varphi_g(x_2),$

where the second equality is given by the insertion of the identity between $x_1$ and $x_2.$ Furthermore, it has a left and right inverse, namely $\varphi_{g^{-1}}.$ Thus, $\varphi_g$ is both an monomorphism and epimorphism, and so an isomorphism of {{mvar|G}} with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.{{Cite book |title=Abstract algebra |first1=David S. |last1=Dummit |first2=Richard M. |last2=Foote |date=2004 |publisher=Wiley |isbn=978-0-4714-5234-8 |edition=3rd |location=Hoboken, NJ |page=45 |oclc=248917264}}

File:Venn Diagram of Homomorphisms.jpg

When discussing right conjugation, the expression $g^{-1}xg$ is often denoted exponentially by $x^g.$ This notation is used because composition of conjugations satisfies the identity: $\left(x^{g_1}\right)^{g_2} = x^{g_1g_2}$ for all $g_1, g_2 \in G.$ This shows that right conjugation gives a right action of {{mvar|G}} on itself.

A common example is as follows:{{Cite book |last=Grillet |first=Pierre |title=Abstract Algebra |publisher=Springer |year=2010 |isbn=978-1-4419-2450-6 |edition=2nd |location=New York |pages=56}}{{Cite book |last=Lang |first=Serge |title=Algebra |publisher=Springer-Verlag |year=2002 |isbn=978-0-387-95385-4 |edition=3rd |location=New York |pages=26}}

File:Diagram of Inn(G) Example.jpg

Describe a homomorphism $\Phi$ for which the image, $\text{Im} (\Phi)$ , is a normal subgroup of inner automorphisms of a group $G$ ; alternatively, describe a natural homomorphism of which the kernel of $\Phi$ is the center of $G$ (all $g \in G$ for which conjugating by them returns the trivial automorphism), in other words, $\text{Ker} (\Phi) = \text{Z}(G)$ . There is always a natural homomorphism $\Phi : G \to \text{Aut}(G)$ , which associates to every $g \in G$ an (inner) automorphism $\varphi_{g}$ in $\text{Aut}(G)$ . Put identically, $\Phi : g \mapsto \varphi_{g}$ .

Let $\varphi_{g}(x) := gxg^{-1}$ as defined above. This requires demonstrating that (1) $\varphi_{g}$ is a homomorphism, (2) $\varphi_{g}$ is also a bijection, (3) $\Phi$ is a homomorphism.

$\varphi_{g}(xx')=gxx'g^{-1} =gx(g^{-1}g)x'g^{-1} = (gxg^{-1})(gx'g^{-1}) = \varphi_{g}(x)\varphi_{g}(x')$
The condition for bijectivity may be verified by simply presenting an inverse such that we can return to $x$ from $gxg^{-1}$ . In this case it is conjugation by $g^{-1}$ denoted as $\varphi_{g^{-1}}$ .
$\Phi(gg')(x)=(gg')x(gg')^{-1}$ and $\Phi(g)\circ \Phi(g')(x)=\Phi(g) \circ (g'xg'^{-1}) = gg'xg'^{-1}g^{-1} = (gg')x(gg')^{-1}$

Inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of {{mvar|G}} is a group, the inner automorphism group of {{mvar|G}} denoted {{math|Inn(G)}}.

{{math|Inn(G)}} is a normal subgroup of the full automorphism group {{math|Aut(G)}} of {{mvar|G}}. The outer automorphism group, {{math|Out(G)}} is the quotient group

: $\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G).$

The outer automorphism group measures, in a sense, how many automorphisms of {{mvar|G}} are not inner. Every non-inner automorphism yields a non-trivial element of {{math|Out(G)}}, but different non-inner automorphisms may yield the same element of {{math|Out(G)}}.

Saying that conjugation of {{mvar|x}} by {{mvar|a}} leaves {{mvar|x}} unchanged is equivalent to saying that {{mvar|a}} and {{mvar|x}} commute:

: $a^{-1}xa = x \iff xa = ax.$

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).

An automorphism of a group {{mvar|G}} is inner if and only if it extends to every group containing {{mvar|G}}.{{Citation|title=A characterization of inner automorphisms|year=1987|last1=Schupp|first1=Paul E.|author-link1=Paul Schupp|journal=Proceedings of the American Mathematical Society|volume=101|issue=2|pages=226–228|publisher=American Mathematical Society|doi=10.2307/2045986|doi-access=free|jstor=2045986|mr=902532|url=https://www.ams.org/journals/proc/1987-101-02/S0002-9939-1987-0902532-X/S0002-9939-1987-0902532-X.pdf}}

By associating the element {{math|a ∈ G}} with the inner automorphism {{math|f(x) {{=}} x{{sup|a}}}} in {{math|Inn(G)}} as above, one obtains an isomorphism between the quotient group {{math|G / Z(G)}} (where {{math|Z(G)}} is the center of {{mvar|G}}) and the inner automorphism group:

: $G\,/\,\mathrm{Z}(G) \cong \operatorname{Inn}(G).$

This is a consequence of the first isomorphism theorem, because {{math|Z(G)}} is precisely the set of those elements of {{mvar|G}} that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).

=Non-inner automorphisms of finite {{mvar|p}}-groups=

A result of Wolfgang Gaschütz says that if {{mvar|G}} is a finite non-abelian p-group, then {{mvar|G}} has an automorphism of {{mvar|p}}-power order which is not inner.

It is an open problem whether every non-abelian {{mvar|p}}-group {{mvar|G}} has an automorphism of order {{mvar|p}}. The latter question has positive answer whenever {{mvar|G}} has one of the following conditions:

{{mvar|G}} is nilpotent of class 2
{{mvar|G}} is a regular p-group
{{math|G / Z(G)}} is a powerful p-group
The centralizer in {{mvar|G}}, {{math|C{{sub|G}}}}, of the center, {{mvar|Z}}, of the Frattini subgroup, {{math|Φ}}, of {{mvar|G}}, {{math|C{{sub|G}} ∘ Z ∘ Φ(G)}}, is not equal to {{math|Φ(G)}}

=Types of groups=

The inner automorphism group of a group {{mvar|G}}, {{math|Inn(G)}}, is trivial (i.e., consists only of the identity element) if and only if {{mvar|G}} is abelian.

The group {{math|Inn(G)}} is cyclic only when it is trivial.

At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called complete. This is the case for all of the symmetric groups on {{mvar|n}} elements when {{mvar|n}} is not 2 or 6. When {{math|n {{=}} 6}}, the symmetric group has a unique non-trivial class of non-inner automorphisms, and when {{math|n {{=}} 2}}, the symmetric group, despite having no non-inner automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete.

If the inner automorphism group of a perfect group {{mvar|G}} is simple, then {{mvar|G}} is called quasisimple.

Lie algebra case

An automorphism of a Lie algebra {{math|𝔊}} is called an inner automorphism if it is of the form {{math|Ad{{sub|g}}}}, where {{math|Ad}} is the adjoint map and {{mvar|g}} is an element of a Lie group whose Lie algebra is {{math|𝔊}}. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.

Extension

If {{mvar|G}} is the group of units of a ring, {{mvar|A}}, then an inner automorphism on {{mvar|G}} can be extended to a mapping on the projective line over a ring by the group of units of the matrix ring, {{math|M{{sub|2}}(A)}}. In particular, the inner automorphisms of the classical groups can be extended in that way.