A note on filled groups

2015/12/16 by Hart, Sarah, Anabanti, Chimere
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1512.05117

Abstract

Let G be a finite group and S a subset of G. Then S is \em product-free if S ∩ SS = ∅, and S \em fills G if G^∗ ⊆ S ∪ SS. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. Street and Whitehead [J. Combin. Theory Ser. A 17 (1974), 219--226] defined a group G as \em filled if every locally maximal product-free set in G fills G. Street and Whitehead 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 the current authors in [Austral. Journal of Combinatorics 63 (3) (2015), 385--398], where we also classified the filled groups of odd order. This brief note completes the classification of filled dihedral groups and discusses filled groups of order up to 100.

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