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

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Statement

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

Facts used

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