# Non-abelian group generated by aperiodic elements has no nontrivial power automorphism

From Groupprops

## Statement

Suppose is a non-abelian group generated by a set all of whose elements are aperiodic, i.e., they have infinite order. Then, has no power automorphisms other than the identity map.

## Facts used

- Cooper's theorem: This states that every power automorphism of a group commutes with every inner automorphism.