Locally Maximal Product-free Sets of Size 3

2015/03/23 by Anabanti, Chimere Stanley, Hart, Sarah
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1503.06509

Abstract

Let G be a group, and S a non-empty subset of G. Then S is product-free if ab∉ S for all a, b ∈ S. We say S is 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| ≤ 24. This allows us to complete the classification of locally maximal product free sets of size 3.

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