Three questions of Bertram on locally maximal sum-free sets

Abstract

Let G be a finite group, and S a sum-free subset of G. The set S is locally maximal in G if S is not properly contained in any other sum-free set in G. If S is a locally maximal sum-free set in a finite abelian group G, then $$G=S\cup SS\cup SS^{-1}\cup \sqrt{S}$$ , where $$SS=\{xy|~x,y\in S\}$$ , $$SS^{-1}=\{xy^{-1}|~x,y\in S\}$$ and $$\sqrt{S}=\{x\in G|~x^2\in S\}$$ . Each set S in a finite group of odd order satisfies $$|\sqrt{S}|=|S|$$ . No such result is known for finite abelian groups of even order in general. In view to understanding locally maximal sum-free sets, Bertram asked the following questions: In this paper, we answer question (i) in the negative, then (ii) and (iii) in affirmative.