Groups Containing Small Locally Maximal Product-Free Sets

Let be a group and a nonempty subset of . Then, is product-free if for all . We say is a locally maximal product-free set if is product-free and not properly contained in any other product-free set. It is natural to ask whether it is possible to determine the smallest possible size of a locally maximal product-free set in . Alternatively, given a positive integer , one can ask the following: what is the largest integer such that there is a group of order with a locally maximal product-free set of size ? The groups containing locally maximal product-free sets of sizes and are known, and it has been conjectured that . 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 .


Introduction
Let  be a group and  a nonempty subset of .Then,  is product-free if  ∉  for all ,  ∈ .For example, if  is a subgroup of  then  is a product-free set for any  ∉ .Traditionally these sets have been studied in abelian groups and have therefore been called sum-free sets (see, e.g., [1,2]).Since we are working with arbitrary groups it makes more sense to say "product-free" in this context.We say  is a locally maximal product-free set if  is product-free and not properly contained in any other product-free set.We use the term locally maximal rather than maximal because the majority of the literature in this area uses maximal to mean maximal by cardinality (e.g., [3,4]).
There are some obvious questions from the definition: given a group , what is the maximum cardinality of a product-free set in  and what are the maximal (by cardinality) product-free sets?How many product-free sets are there in ?Given that each product-free set is contained in a locally maximal product-free set, what are the locally maximal product-free sets?What are the possible sizes of locally maximal product-free sets?Most of the work on product-free sets has been done in the abelian group case, particularly for Z and Z  .The number of sum-free sets in the integers has been studied by, for example, Cameron and Erdös [1] and Green [5], who with Ruzsa also studied the density and number of sum-free sets in abelian groups [6].The number of sum-free sets of {1, 2, . . ., } is 2 (1/2+(1)) .The number of sum-free sets of an arbitrary abelian group is 2 (()+(1)) .See [7] for further work in this direction.Petrosyan [8] determined the asymptotic behaviour of the number of product-free sets in groups of even order.Green and Ruzsa in [6] also determined the maximal size of a sumfree set in an arbitrary abelian group.For the nonabelian case Kedlaya [9] showed that there exists a constant  such that the largest product-free set in a group of order  is of size at least  11/14 .See also [10].Gowers in his work on quasirandom groups proved that if the smallest nontrivial representation of a group  is of dimension  then the largest product-free set in  is of size at most  −1/3  (Theorem 3.3 and commentary at the start of Section 5 of [11]).Much less is known about the minimum sizes of locally maximal productfree sets.This question was first asked in [3] and later in [12], where the authors ask what is the minimum size of a locally maximal product-free set in a group of order ?A good bound for this minimum size is still not known.Small locally maximal product-free sets when  is an elementary abelian 2-group are of interest in finite geometry, because they correspond to complete caps in PG(−1, 2).Locally maximal sum-free sets for elementary abelian 2-groups of order up to 64 were classified in [13].In [14], all groups containing locally 2 International Journal of Combinatorics maximal product-free sets of sizes 1 and 2 were classified.Some general results were also obtained.Furthermore, there was a classification (Theorem 5.6 of [14]) of groups containing locally maximal product-free sets  of size 3 for which not every subset of size 2 in  generates ⟨⟩.Each of these groups has order of at most 24.Conjecture 5.7 of [14] was that if  is a group of order greater than 24, then  does not contain a locally maximal product-free set of size 3.A list was given of all locally maximal product-free sets in groups of orders up to 24.So the conjecture asserts that this list is the complete list of all such sets.(This list is given in the current paper as Table 1; we include it both for ease of reference and because information from it is needed in the proofs of our results.)The main result of this paper is the following and its immediate corollary.
Theorem 1. Suppose  is a locally maximal product-free set of size 3 in a group , such that every two-element subset of  generates ⟨⟩.Then || ≤ 24.
Corollary 2. If a group  contains a locally maximal productfree set  of size 3, then || ≤ 24 and the only possibilities for  and  are listed in Table 1.
Proof.If not every two-element subset of  generates ⟨⟩, then, by Theorem 5.6 of [14], || ≤ 24.We may therefore assume that every two-element subset of  generates ⟨⟩.Then || ≤ 24 by Theorem 1.Now Table 1 is a list of all locally maximal product-free sets of size 3 occurring in groups of order up to 24 (a version of this table appeared in [14] as a list of such sets in groups of order up to 37).Since we have shown that all locally maximal product-free sets of size 3 occur in groups of order up to 24, this table now constitutes a complete list of possibilities.
More generally, given a positive integer , one can ask the following: what is the largest integer   such that there is a group of order   with a locally maximal product-free set of size ?Using GAP [15] we have tested all groups of order up to 100 when  ≤ 5, and the results suggest that the sequence   begins 8, 16, 24, 40, 64, which means the sequence (1/8)  begins 1, 2, 3, 5, 8.This is rather intriguing and it would be interesting to know what the sequence actually is.
We finish this section by establishing the notation to be used in the rest of the paper and giving some basic results from [14].For subsets  and  of a group , we use the standard notation  for the product of  and .That is,  = { :  ∈ ,  ∈ }.By definition, a nonempty set  ⊆  is product-free if and only if ∩ = ⌀.In order to investigate locally maximal product-free sets, we introduce some further notations.For a set  ⊆ , we define the following sets: For a singleton set {}, we usually write √ instead of √{}.
For a positive integer , we will denote by Alt() the alternating group of degree , by   the cyclic group of order , by  2 the dihedral group of order 2, and by  4 the dicyclic group of order 4 given by  4 := ⟨,  :  2 = 1,   =  2 ,  =  −1 ⟩.
We now state the results from [14] that we will use.
We require one final fact.
Theorem 5 (see [14,Theorem 5.1]).Up to isomorphism, the only instances of locally maximal product-free sets  of size 3 of a group  where || ≤ 37 are given in Table 1.

Proof of Theorem 1
Proposition 6. Suppose  is a locally maximal product-free set of size 3 in .If ⟨⟩ is cyclic, then || ≤ 24.
Proof.Write  = {, , }.First note that since ⟨⟩ is abelian,   It remains to consider  6 ,  8 ,  10 , and  12 .For  6 = ⟨ :  6 = 1⟩, the unique locally maximal product-free set of size 3 is  = {,  3 ,  5 }.Now if  or  5 is contained in Ŝ, then Ŝ consists of powers of a single element; so, by Theorem 4(v), || divides 24.If neither  nor  5 is in Ŝ, then | Ŝ| ≤ 1, and so by Theorem 4(iii), therefore, || divides 12.In  8 there is a unique (up to group automorphisms) locally maximal product-free set of size 3, and it is {,  −1 ,  4 }, where  is any element of order 8.If Ŝ contains  or  −1 , then  contains all odd powers of that element by Theorem 4(iv), and hence  contains {,  3 ,  5 ,  7 }, a contradiction.Therefore | Ŝ| ≤ 1 and so || divides 16.Next, we consider ⟨⟩ =  10 .Recall that elements of Ŝ must have even order.If Ŝ contains any element of order 10, then  contains all five odd powers of this element, which is impossible by Theorem 4(iv).This leaves only the involution of  10 as a possible element of Ŝ. Hence again | Ŝ| ≤ 1 and || divides 20.Finally we look at  12 .If Ŝ contains any element of order 12, then || ≥ 6, a contradiction.If Ŝ contains an element  of order 6 then  contains all three of its odd powers, so  = {,  3 ,  5 }.But then ⟨⟩ ≅  6 , contradicting the assumption that ⟨⟩ =  12 .Therefore, Ŝ can only contain elements of order 2 or 4. Up to group automorphism, we see from Table 1 that every locally maximal product-free set  of size 3 in  12 with ⟨⟩ =  12 is one of {,  6 ,  10 } or {,  3 ,  8 } for some generator  of  12 .Each of these sets contains exactly one element of order 2 or 4. Therefore, in every case, | Ŝ| ≤ 1 and so || divides 24.This completes the proof.
Note that the bound on || in Proposition 6 is attainable.For example, in  24 there is a locally maximal product-free set  of size 3, with ⟨⟩ ≅  12 .Proposition 7. Suppose  is a locally maximal product-free set of size 3 in  such that every 2-element subset of  generates ⟨⟩.Then either || ≤ 24 or  contains exactly one involution.
(4) Lemma 8. Suppose  is a locally maximal product-free set of size 3 in , every 2-element subset of  generates ⟨⟩, and  contains exactly one involution.Then either || ≤ 24 or  = {, , }, where ,  have order 3 and  is an involution.
We can now prove Theorem 1, which states that if  is a locally maximal product-free set of size 3 in a group , such that every two-element subset of  generates ⟨⟩, then || ≤ 24.

A Table of All Locally Maximal
Product-Free Sets of Size 3 Though Table 1 is essentially the same as the one in [14], we have taken the opportunity here to correct a typographical error in the entry for the (unnamed) group of order 16.
The data was obtained using simple GAP programs [15] and additionally verified by hand for the smaller groups.