Inner automorphism

In abstract algebra an inner automorphism is a certain type of automorphism of a group defined in terms of a fixed element of the group, called the conjugating element. Formally, if G is a group and a is an element of G, then the inner automorphism defined by a is the map f from G to itself defined for all $x$ in $G$ by the formula

f (x) = a -1 xa

Here we use the convention that group elements act on the right.

The operation $x \mapsto a -1 xa$ is called conjugation (see also conjugacy class), and it is often of interest to distinguish the cases where conjugation by one element leaves another element unchanged from cases where conjugation generates a new element.

In fact, saying that conjugation of $x$ by $a$ leaves $x$ unchanged is equivalent to saying that $a$ and $x$ commute:

a -1 xa = x \Leftrightarrow ax = xa

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.

An automorphism of a group $G$ is inner if and only if it extends to every group containing $G$ .^[1]

Notation

The expression $a -1 xa$ is often denoted exponentially by $x a$ . This notation is used because we have the rule $(x a) b = x ab$ (giving a right action of $G$ on itself).

Properties

Every inner automorphism is indeed an automorphism of the group $G$ , i.e. it is a bijective map from $G$ to $G$ and it is a homomorphism; meaning that $(xy) a = x a y a$ .

Inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism (as mentioned above: $(x a) b = x ab$ ), and with this operation, the collection of all inner automorphisms of $G$ is itself a group, the inner automorphism group of $G$ denoted $Inn(G)$ .

$Inn(G)$ is a normal subgroup of the full automorphism group $Aut(G)$ of $G$ . The outer automorphism group, $Out(G)$ is the quotient group

Out(G) \equiv Aut(G)/Inn(G)

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

By associating the element $a \in G$ with the inner automorphism $f (x) = x a$ in $Inn(G)$ as above, one obtains an isomorphism between the quotient group $G /Z(G)$ (where $Z(G)$ is the centre of $G$ ) and the inner automorphism group:

G /Z(G) = Inn(G)

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

Non-inner automorphisms of finite $p$ -groups

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

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

$G$ is nilpotent of class 2
$G$ is a regular $p$ -group
$G /Z(G)$ is a powerful $p$ -group
The centralizer in $G$ , $C G$ , of the centre, $Z$ , of the Frattini subgroup, $Φ$ , of $G$ , $C G \circZ\circΦ(G)$ , is not equal to $Φ(G)$

Types of groups

It follows that the inner automorphism group, $Inn(G)$ , is trivial (i.e., consists only of the identity element) if and only if $G$ is abelian.

It is easy to prove that $Inn(G)$ can be a cyclic group 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 centre is trivial is called complete. This is the case for all of the symmetric groups on $n$ elements when $n$ is not 2 or 6, when $n = 6$ the symmetric group has a unique non-trivial class of outer automorphisms, and when $n = 2$ the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial centre disqualifying it from being complete.

If the inner automorphism group of a perfect group $G$ is simple, then $G$ is called quasisimple.

Ring case

Given a ring, $R$ , and a unit, $u$ in $R$ , the map $f (x) = u -1 xu$ is a ring automorphism of $R$ . The ring automorphisms of this form are called inner automorphisms of $R$ . They form a normal subgroup of the automorphism group of $R$ .

Lie algebra case

An automorphism of a Lie algebra $𝔊$ is called an inner automorphism if it is of the form $Ad g$ , where $Ad$ is the adjoint map and $g$ is an element of a Lie group whose Lie algebra is $𝔊$ . 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 $G$ is the group of units of a ring, $A$ , then an inner automorphism on $G$ can be extended to a mapping on the projective line over $A$ by the group of units of the matrix ring, $M 2 (A)$ . In particular, the inner automorphisms of the classical groups can be extended in that way.

References

↑ Schupp, Paul E. (1987), "A characterization of inner automorphisms", Proceedings of the American Mathematical Society, American Mathematical Society, 101 (2): 226–228, JSTOR 2045986, MR 902532, doi:10.2307/2045986