On a conjecture of Street and Whitehead on locally maximal product-free sets

TL;DR

This paper investigates filled groups and locally maximal product-free sets, disproves a conjecture about dihedral groups, and classifies such sets in dihedral groups of certain orders, extending understanding beyond abelian groups.

Contribution

It disproves Street and Whitehead's conjecture on dihedral groups and classifies locally maximal product-free sets of sizes 3 and 4 in dihedral groups.

Findings

Disproved the conjecture that dihedral groups of order 2(6k+1) are not filled.

Classified filled groups of odd order.

Characterized locally maximal product-free sets of sizes 3 and 4 in dihedral groups.

Abstract

Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is product-free and $S\subseteq T$, then $S=T$. Finally $S$ fills $G$ if $G^*\subseteq S \sqcup SS$ (where $G^*$ is the set of all non-identity elements of $G$), and $G$ is a filled group if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead (in `Group Ramsey Theory', J. Comb. Theory Series A, 17 (1974) 219-226) investigated filled groups and gave a classification of filled abelian groups. In this paper, we obtain some results about filled groups in the non-abelian case, including a classification of filled groups of odd order. Street and Whitehead conjectured that the finite dihedral group of order $2n$ is not filled when $n=6k+1$ ($k\geq 1$). We disprove this conjecture on dihedral groups, and in doing so obtain a classification of locally maximal product-free sets of sizes 3 and 4 in dihedral groups.