An inner automorphism of a group is an automorphism of the form , where is a fixed element of . An outer automorphism of is an automorphism which cannot be expressed in this form for , but can be so expressed if belongs to a larger group containing .

For example, the automorphism of the symmetric group which maps the permutation to is an inner automorphism, since . However, it is an outer automorphism of the alternating group since does not belong to and there is no element of such that .