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In mathematics, the outer automorphism group of a group, G, is the quotient, Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out. If Out is trivial and G has a trivial center, then G is said to be complete.
An automorphism of a group which is not inner is called an outer automorphism. The cosets of Inn with respect to outer automorphisms are then the elements of Out; this is an instance of the fact that quotients of groups are not, in general, subgroups. If the inner automorphism group is trivial , the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group.
For example, for the alternating group, An, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group, Sn, conjugation by any odd permutation is an outer automorphism of An or more precisely "represents the class of the outer automorphism of An", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.