Small maximal sum-free sets

Abstract

Let G be a group and S a non-empty subset of G. If ab∉S for any a,b∈S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates ⟨S⟩ then |S|≤2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a∈S such that a∉⟨S∖{a}⟩