Groups whose locally maximal product-free sets are complete

Abstract

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is product-free if $S \cap SS = \emptyset$, and complete if $G^{\ast} \subseteq S \cup SS$. 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 [J. Combin. Theory Ser. A 17 (1974), 219--226] 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 $n=6k+1$ ($k\geq 1$). The conjecture was disproved by two of the current authors in [Austral. J. Combin. 63 (3) (2015), 385--398], 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 $2^np$ where $p$ is an odd prime. We use these results to determine all filled groups of order up to 2000.