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

Groups containing locally maximal product free sets of size 4

Every locally maximal product-free set S in a finite group G satisfies G = S[SS[S−1S[ SS−1 [pS, where SS = {xy| x, y 2 S}, S−1S = {x−1y| x, y 2 S}, SS−1 = {xy−1| x, y 2 S} and pS = {x 2 G| x2 2 S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |pS| � 2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case

A note on filled groups

Let G be a finite group and S a subset of G. Then S is product-free if S \ SS = ;, and S 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 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

Groups whose locally maximal product-free sets are complete

Let G be a finite group and S a subset of G. Then S is product-free if S ∩ SS = ∅, and complete if G∗ ⊆ S ∪ 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 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

Groups containing small locally maximal product-free sets

Let G be a group, and S a non-empty subset of G. Then S is product-free if ab is not in S for all a, b in S. We say S is locally maximal product-free if S is product-free and not properly contained in any other product-free set. A natural question is to determine the smallest possible size of a locally maximal product-free set in G. Alternatively, given a positive integer k, one can ask: what is the largest integer n_k such that there is a group of order n_k with a locally maximal product-free set of size k? The groups containing locally maximal product-free sets of sizes 1 and 2 are known, and it has been conjectured that n_3 = 24. The purpose of this paper is to prove this conjecture and hence show that the list of known locally maximal product-free sets of size 3 is complete. We also report some experimental observations about the sequence n_k

Locally maximal product-free sets of size 3

Let G be a group, and S a non-empty subset of G. Then S is product-free if ab =2 S for all a; b 2 S. We say S is locally maximal product-free if S is product-free and not properly contained in any other product-free set. A natural question is what is the smallest possible size of a locally maximal product-free set in G. The groups containing locally maximal product-free sets of sizes 1 and 2 were classi�ed in [3]. In this paper, we prove a conjecture of Giudici and Hart in [3] by showing that if S is a locally maximal product-free set of size 3 in a group G, then jGj � 24. This shows that the list of known locally maximal product-free sets given in [3] is complete

Three questions of Bertram on locally maximal sum-free sets

Let G be a finite group, and S a sum-free subset of G. The set S is locally maximal in G if S is not properly contained in any other sum-free set in G. If S is a locally maximal sum-free set in a finite abelian group G, then G = S [ SS [ SS−1 [ pS, where SS = {xy| x, y 2 S}, SS−1 = {xy−1| x, y 2 S} and pS = {x 2 G| x2 2 S}. Each set S in a finite group of odd order satisfies |pS| = |S|. No such result is known for finite abelian groups of even order in general. In view to understanding locally maximal sum-free sets, Bertram asked the following questions: (i) Does S locally maximal sum-free in a finite abelian group imply |pS| � 2|S|? (ii) Does there exists a sequence of finite abelian groups G and locally maximal sum-free sets S � G such that |SS| |S| ! 1 as |G| ! 1? (iii) Does there exists a sequence of abelian groups G and locally maximal sum-free sets S � G such that |S| < c|G|1 2 as |G| ! 1, where c is a constant? In this paper, we answer question (i) in the negation, then (ii) and (iii) in affirmation

Groups whose locally maximal product-free sets are complete

Let G be a finite group and S a subset of G. Then S is product-free if S \ SS = ;, and complete if G� � S [ 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 [11] 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 in [2], 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 2np where p is an odd prime. We use these results to determine all filled groups of order up to 2000