# Inner automorphism

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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. These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup gives rise to the concept of the outer automorphism group.

## Definition

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

{\begin{aligned}\varphi _{g}\colon G&\longrightarrow G\\\varphi _{g}(x)&:=g^{-1}xg\end{aligned}} is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all $x_{1},x_{2}\in G,$ $\varphi _{g}(x_{1}x_{2})=g^{-1}x_{1}x_{2}g=(g^{-1}x_{1}g)(g^{-1}x_{2}g)=\varphi _{g}(x_{1})\varphi _{g}(x_{2}),$ 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 bijective, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.

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: $(x^{g_{1}})^{g_{2}}=x^{g_{1}g_{2}}$ for all $g_{1},g_{2}\in G.$ This shows that conjugation gives a right action of G on itself.

## 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 G is 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

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

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

$a^{-1}xa=x\iff ax=xa.$ 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 (or ring).

An automorphism of a group G is inner if and only if it extends to every group containing G.

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

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

1. G is nilpotent of class 2
2. G is a regular p-group
3. G/Z(G) is a powerful p-group
4. The centralizer in G, CG, of the center, Z, of the Frattini subgroup, Φ, of G, CGZ ∘ Φ(G), is not equal to Φ(G)

### Types of groups

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

The group 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 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 center, disqualifying it from being complete.

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

## Lie algebra case

An automorphism of a Lie algebra 𝔊 is called an inner automorphism if it is of the form Adg, 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, M2(A). In particular, the inner automorphisms of the classical groups can be extended in that way.