Groups with an automorphism cubing many elements

Abstract

Let G be a group and αn the mapping which takes every element of G to its nth power, where n is an integer. It is well known that if αn is an automorphism then G is Abelian in the cases n = -1,2, and 3. For any other integer n(≠ 0) there exists a non-Abelian group which admits αn as the identity automorphism. Indeed Miller (1929) has shown that if n ≠ 0, ±1, 2, 3 then there exist non-Abelian groups which admit αn as a non-trivial automorphism.