Small maximal sum-free sets

Let G be a group and S a non-empty subset of G. If ab /∈ S for any a, b ∈ S, then Sis called sum-free. We show that if S is maximal by inclusion and no proper subsetgenerates hSi 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 thata /∈ hS \ {a}i.