Groups Containing Small Locally Maximal Product-Free Sets

Chimere S Anabanti,Sarah B Hart

doi:10.1155/2016/8939182

Abstract

Let G be a group and S a nonempty subset of G. Then, S is product-free if ab∉S for all a,b∈S. We say S is a locally maximal product-free set if S is product-free and not properly contained in any other product-free set. It is natural to ask whether it is possible to determine the smallest possible size of a locally maximal product-free set in G. Alternatively, given a positive integer k, one can ask the following: what is the largest integer nk such that there is a group of order nk with a locally maximal product-free set of size k? The groups containing locally maximal product-free sets of sizes 1 and 2 are known, and it has been conjectured that n3=24. The purpose of this paper is to prove this conjecture and hence show that the list of known locally maximal product-free sets of size 3 is complete. We also report some experimental observations about the sequence nk.

Highlights

Let G be a group and S a nonempty subset of G
Suppose S is a locally maximal product-free set of size 3 in a group G, such that every two-element subset of S generates ⟨S⟩
Since we have shown that all locally maximal product-free sets of size 3 occur in groups of order up to 24, this table constitutes a complete list of possibilities

Summary

Introduction

Let G be a group and S a nonempty subset of G. There was a classification (Theorem 5.6 of [14]) of groups containing locally maximal product-free sets S of size 3 for which not every subset of size 2 in S generates ⟨S⟩ Each of these groups has order of at most 24. Given a positive integer k, one can ask the following: what is the largest integer nk such that there is a group of order nk with a locally maximal product-free set of size k?

Proof of Theorem 1

Findings

A Table of All Locally Maximal Product-Free Sets of Size 3