Editing Outer automorphism group (section)

{{Short description|Mathematical group}}
In [[mathematics]], the '''outer automorphism group''' of a [[group (mathematics)|group]], {{mvar|G}}, is the [[quotient group|quotient]], {{math|Aut(''G'')&nbsp;/&nbsp;Inn(''G'')}}, where {{math|Aut(''G'')}} is the [[automorphism group]] of {{mvar|G}} and {{math|Inn(''G''}}) is the subgroup consisting of [[inner automorphism]]s. The outer automorphism group is usually denoted {{math|Out(''G'')}}. If {{math|Out(''G'')}} is trivial and {{mvar|G}} has a trivial [[Center (group theory)|center]], then {{mvar|G}} is said to be [[complete group|complete]].

An automorphism of a group that is not inner is called an '''outer automorphism'''.<ref>Despite the name, these do not form the elements of the outer automorphism group.  For this reason, the term ''non-inner automorphism'' is sometimes preferred.</ref> The [[coset|cosets]] of {{math|Inn(''G'')}} with respect to outer automorphisms are then the elements of {{math|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]], {{math|A{{sub|''n''}}}}, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering {{math|A{{sub|''n''}}}} as a subgroup of the [[symmetric group]], {{math|S{{sub|''n''}}}}, conjugation by any [[odd permutation]] is an outer automorphism of {{math|A{{sub|''n''}}}} or more precisely "represents the class of the (non-trivial) outer automorphism of {{math|A{{sub|''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.