Outer automorphism group

In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. An automorphism of a group that is not inner is called an outer automorphism. The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), 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, A_n, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering A_n as a subgroup of the symmetric group, S_n, conjugation by any odd permutation is an outer automorphism of A_n or more precisely "represents the class of the (non-trivial) outer automorphism of A_n", 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. The Schreier conjecture asserts that Out(G) is always a solvable group when G is a finite simple group. This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known. The outer automorphism group is dual to the center in the following sense: conjugation by an element of G is an automorphism, yielding a map σ : G → Aut(G). The kernel of the conjugation map is the center, while the cokernel is the outer automorphism group (and the image is the inner automorphism group). This can be summarized by the exact sequence: Z(G) ↪ G Aut(G) ↠ Out(G). The outer automorphism group of a group acts on conjugacy classes, and accordingly on the character table.