Groups whose locally maximal product - free sets are complete

Let G be a finite group and S a subset of G. Then S is product-free if S ∩ SS = ∅, and complete if G∗ ⊆ S ∪ SS. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. If S is product-free and complete then S is locally maximal, but the converse does not necessarily hold. Street and Whitehead [11] defined a group G as filled if every locally maximal product-free set S in G is complete (the term comes from their use of the phrase ‘S fills G’ to mean S is complete). They classified all abelian filled groups, and conjectured that the finite dihedral group of order 2n is not filled when n = 6k + 1 (k ≥ 1). The conjecture was disproved by two of the current authors in [2], where we also classified the filled groups of odd order. In this paper we classify filled dihedral groups, filled nilpotent groups and filled groups of order 2np where p is an odd prime. We use these results to determine all filled groups of order up to 2000.