Triple-product-free sets (Q6666499)

Let \(G\) be a group. A non-empty subset \(S \subseteq G\) is triple-product-free if \(abc \not \in S\) for all \(a, b, c \in S\). Further, \(S\) is said to be locally maximal triple-product-free if \(S\) is triple-product-free and not properly contained in any other triple-product-free set.\N\NIn the paper under review, the authors classify all groups containing a locally maximal triple-product-free set of size 1. Theorem 2.2: Let \(S\) be a locally maximal triple-product-free set of size 1 in a group \(G\). Then \(G\) is isomorphic to one of \(C_{3}\), \(C_{4}\), \(C_{5}\), \(D_{6}\), \(D_{8}\), \(Q_{8}\) or \(D_{10}\). In Theorem 3.5, they derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free.\N\NIt is interesting to note that the largest possible orders of a group containing a locally maximal product-free set of size \(k\), for \(k = 1, 2, 3, 4\) respectively are \(8, 16, 24, 40\), with computer experiment strongly suggesting 64 for \(k=5\) (see the first author's PhD thesis, 2022).