Groups whose locally maximal product-free sets are complete

Anabanti, C. and Erskine, G. and Hart, Sarah (2018) Groups whose locally maximal product-free sets are complete. Australasian Journal of Combinatorics 71 (3), pp. 544-563. ISSN 2202-3518.

Abstract

Let G be a finite group and S a subset of G. Then S is product-free if $S \cap SS = \emptyset$, and complete} if $G^{\ast} \subseteq S \cup SS$. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. If S is product-free and complete then S is locally maximal, but the converse does not necessarily hold. Street and Whitehead (J. Combin. Theory Ser. A \textbf{17} (1974), 219--226) defined a group G as filled if every locally maximal product-free set S in G is complete (the term comes from their use of the phrase "S fills G" to mean S is complete). They classified all abelian filled groups, and conjectured that the finite dihedral group of order 2n is not filled when n=6k+1 (k\geq 1). The conjecture was disproved by two of the current authors in Austral. Journal of Combinatorics 63 (3) (2015), 385--398, where we also classified the filled groups of odd order. In this paper we classify filled dihedral groups, filled nilpotent groups and filled groups of order 2^np where p is an odd prime. We use these results to determine all filled groups of order up to 2000.

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