Groups in which Raising to a Power is an Automorphism

Extract

For any group G and integer n, let P_n:G→G be the function defined by P_n(g) = gⁿ for all g ϵ G. If G is abelian then P_n is a homomorphism for all n. Conversely, it is well known (and easy to show) that if P₂ or P_-1 is a homomorphism then G is abelian. As the groups G_n described below show, for every n other than 2 and -1 there exist non-abelian groups for which P_n is a homomorphism.