# Inner automorphism

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 simple 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 gives rise to the concept of the outer automorphism group.

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

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

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:

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.^{[2]}

By associating the element *a* ∈ *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 center of G) and the inner automorphism group:

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).

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:

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.

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.

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.

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.