Locally Maximal Product-free Sets of Size 3

TL;DR

This paper proves that any group containing a locally maximal product-free set of size 3 must have order at most 24, completing the classification of such sets of size 3.

Contribution

It confirms a conjecture that groups with a size 3 locally maximal product-free set have order at most 24, completing the classification for size 3 sets.

Findings

Groups with size 3 locally maximal product-free sets have order ≤ 24

Classification of such sets is now complete for size 3

Extends previous classifications for sizes 1 and 2

Abstract

Let $G$ be a group, and $S$ a non-empty subset of $G$. Then $S$ is \emph{product-free} if $ab\notin S$ for all $a, b \in S$. We say $S$ is \emph{locally maximal product-free} if $S$ is product-free and not properly contained in any other product-free set. A natural question is what is the smallest possible size of a locally maximal product-free set in $G$. The groups containing locally maximal product-free sets of sizes $1$ and $2$ were classified by Giudici and Hart in 2009. In this paper, we prove a conjecture of Giudici and Hart by showing that if $S$ is a locally maximal product-free set of size $3$ in a group $G$, then $|G| \leq 24$. This allows us to complete the classification of locally maximal product free sets of size 3.