THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS

Abstract

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.