# Extended automorphism group

## 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 generated by the automorphism group and the inverse map. See every group is isomorphic to its opposite group via the inverse map and inverse map is involutive. Moreover the automorphism group and the inverse map commute. Thus, there are two cases:

• The inverse map is itself an automorphism. This happens if and only if the group is an abelian group, in which case the extended automorphism group equals the automorphism group. Further information: inverse map is automorphism iff abelian
• The inverse map is not an automorphism. This happens if and only if the group is a non-abelian group. In this case, the extended automorphism group is the internal direct product of the automorphism group and the cyclic group of order two generated by the inverse map.