A Structure Theorem for Small Sumsets in Nonabelian Groups

TL;DR

This paper extends classical sumset structure theorems from abelian to nonabelian groups, showing that small sumsets are either geometric progressions or closely related to subgroup cosets, with detailed counterexamples.

Contribution

It provides a nonabelian generalization of sumset structure theorems and characterizes the structure of small sumsets with and without size restrictions.

Findings

Small sumsets are either geometric progressions or related to subgroup cosets.

Counterexamples exist when size restrictions are removed.

The structure theorem extends classical abelian results to nonabelian groups.

Abstract

Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.