Table of Contents
International Journal of Combinatorics
Volume 2016 (2016), Article ID 8939182, 5 pages
http://dx.doi.org/10.1155/2016/8939182
Research Article

Groups Containing Small Locally Maximal Product-Free Sets

Birkbeck, University of London, London, UK

Received 19 July 2016; Accepted 15 September 2016

Academic Editor: Laszlo A. Szekely

Copyright © 2016 Chimere S. Anabanti and Sarah B. Hart. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a group and a nonempty subset of . Then, is product-free if for all . We say is a locally maximal product-free set if 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 . Alternatively, given a positive integer , one can ask the following: what is the largest integer such that there is a group of order with a locally maximal product-free set of size ? The groups containing locally maximal product-free sets of sizes and are known, and it has been conjectured that . 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 .