Groups whose locally maximal product-free sets are complete

Anabanti, Chimere S.; Erskine, Grahame and Hart, Sarah B. (2018). Groups whose locally maximal product-free sets are complete. Australasian Journal of Combinatorics, 71(3) pp. 544–563.



Let G be a finite group and S a subset of G. Then S is product-free if SSS = ∅, and complete if G∗ ⊆ SSS. 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 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 ≥ 1). The conjecture was disproved by two of the current authors [C.S. Anabanti and S.B. Hart, Australas. J. Combin. 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 2n p where p is an odd prime. We use these results to determine all filled groups of order up to 2000.

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