Power-translation invariance in discrete groups—A characterization of finite Hamiltonian groups

The primary result of this paper is the resolution of the question: Which non-Abelian discrete groups G satisfy (for some n>1) |(aS)ⁿ|=|Sⁿ| for all S G and a∈G, (A_n) where |*| denotes the counting measure and Sⁿ={s₁…s_n:s_i∈S, 1≤i≤n}?

We prove that a discrete group G satisfies (A_n) for some integer n>1 iff G is a finite Hamiltonian group. Furthermore, if γ denotes the invariant defined for finite Abelian groups introduced in [1] and H is any finite Hamiltonian group, then H satisfies (A_n) iff γ(H′)≥n, where H′ denotes the unique (up to isomorphism) maximal Abelian subgroup of H. In the course of this development a number of results concerning finite Hamiltonian groups are obtained. We conclude with a section on related conditions as well as a discussion of the general locally compact case.