On a conjecture of Street and Whitehead on locally maximal product-free sets

Let S be a non-empty subset of a group G. We say S is product-free if S contains no solutions to ab=c, and S is locally maximal if whenever T is product-free and S is a subset of T, then S = T. Finally S fills G if every non-identity element of G is contained either in S or SS, and G is a filled group if every locally maximal product-free set in G fills G. Street and Whitehead [J. Combin. Theory Ser. A 17 (1974), 219–226] investigated filled groups and gave a classification of filled abelian groups. In this paper, we obtain some results about filled groups in the non-abelian case, including a classification of filled groups of odd order. Street and Whitehead conjectured that the finite dihedral group of order 2n is not filled when n = 6k + 1 (k a positive integer). We disprove this conjecture on dihedral groups, and in doing so obtain a classification of locally maximal product-free sets of sizes 3 and 4 in dihedral groups