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).
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:
An automorphism of a group G is inner if and only if it extends to every group containing G.
By associating the element a ∈ G 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:
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:
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.
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.
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.