Extended automorphism group

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Definition

Let G be a group. The extended automorphism group of G is the group comprising all the automorphisms and anti-automorphisms of G. An anti-automorphism is a map \varphi:G \to G such that \varphi(gh) = \varphi(h)\varphi(g) for all g,h \in G.

The extended automorphism group is either equal to the automorphism group or contains the automorphism group as a subgroup of index two. Equality holds if and only if the group is an abelian group.