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 which is not inner is called an outer automorphism. Note that the elements of Out(G) are cosets of automorphisms of G, and not themselves automorphisms; this is an instance of the fact that quotients of groups are not in general (isomorphic to) subgroups. Elements of Out(G) are cosets of Inn(G) in Aut(G).
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 (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.
However, for an abelian group A, the inner automorphism group is trivial and thus the automorphism group and outer automorphism group are naturally identified, and outer automorphisms do act on A.
Read more about Outer Automorphism Group: Out(G) For Some Finite Groups, The Outer Automorphisms of The Symmetric and Alternating Groups, Outer Automorphism Groups of Complex Lie Groups, Outer Automorphism Groups of Complex and Real Simple Lie Algebras, Structure, Dual To Center, Applications, Puns
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