# Antiautomorphism of a group

From Groupprops

*This article defines a function property, viz a property of functions from a group to itself*

## Definition

An **antiautomorphism** of a group is a map from the group to itself that is an antihomomorphism, and whose inverse is also an antihomomorphism.

The set of automorphisms and antiautomorphisms of a (nontrivial) group together form a group, and the group of automorphisms is a subgroup of index two in this group. The subgroup is a complemented normal subgroup, with the complement being the two-element group comprising the identity map and the inverse map.