Small maximal sum-free sets

Let G be a group and S a non-empty subset of G. If ab∉S for any a,b∈S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates ⟨S⟩ then |S|≤2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a∈S such that a∉⟨S∖{a}⟩

On sum-free subsets of abelian groups

In this paper we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind, and is hence not overly technical. We consider answers to questions like: how many sum-free subsets are there in a given abelian group $G$? what are its sum-free subsets of maximum cardinality? what is the maximum cardinality of these sum-free subsets? what does a typical sum-free subset of $G$ looks like? among others

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

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

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

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

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

On Sum-Free Subsets of Abelian Groups

In this paper, we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind and is hence not overly technical. We consider answers to questions like the following: How many sum-free subsets are there in a given abelian group G? What are its sum-free subsets of maximum cardinality? What is the maximum cardinality of these sum-free subsets? What does a typical sum-free subset of G look like

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