A note on filled groups

TL;DR

This paper completes the classification of filled dihedral groups, building on prior work that classified odd order filled groups and disproved a conjecture about certain dihedral groups, and also discusses filled groups up to order 100.

Contribution

It finalizes the classification of filled dihedral groups and provides insights into filled groups of small order, extending previous classifications and disproving a conjecture.

Findings

Classified all filled dihedral groups.

Disproved the conjecture that certain dihedral groups are not filled.

Analyzed filled groups of order up to 100.

Abstract

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is {\em product-free} if $S \cap SS = \emptyset$, and $S$ {\em fills} $G$ 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. Street and Whitehead [J. Combin. Theory Ser. A \textbf{17} (1974), 219--226] defined a group $G$ as {\em filled} if every locally maximal product-free set in $G$ fills $G$. Street and Whitehead 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 the current authors in [Austral. Journal of Combinatorics \textbf{63 (3)} (2015), 385--398], where we also classified the filled groups of odd order. This brief note completes the classification of filled dihedral groups and discusses filled groups of order up to 100.