Outer automorphism group

[1] 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.

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 (non-trivial) 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.

This result is now known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.

These extensions are not always semidirect products, as the case of the alternating group A6 shows; a precise criterion for this to happen was given in 2003.

Note that, in the case of G = A6 = PSL(2, 9), the sequence 1 ⟶ G ⟶ Aut(G) ⟶ Out(G) ⟶ 1 does not split.

Let G now be a connected reductive group over an algebraically closed field.

Associated to the Borel subgroup is a set of simple roots, and the outer automorphism may permute them, while preserving the structure of the associated Dynkin diagram.

In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G).