Small Maximal Sum-Free Sets

Abstract

Let $G$ be a group and $S$ a non-empty subset of $G$. If $ab \notin S$ for any $a, b \in S$, then $S$ is called sum-free. We show that if $S$ is maximal by inclusion and no proper subset generates $\langle S\rangle$ then $|S|\leq 2$. 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 $a \in S$ such that $a \notin \langle S \setminus \{a\}\rangle$.

Three questions of Bertram on locally maximal sum-free sets

“…Among other results, they calculated locally maximal sum-free sets in groups of small orders, up to 16 in [18,19] as well as a few higher sizes. Going in another direction, Giudici and Hart [13] started the classification of finite groups containing locally maximal sum-free sets. They classified all finite groups containing locally maximal sum-free sets of sizes 1 and 2, as well as some of size 3.…”

“…where SS = {xy| x, y ∈ S}, SS −1 = {xy −1 | x, y ∈ S} and √ S = {x ∈ G| x 2 ∈ S} (see [13,Lemma 3.1]). Each (locally maximal sum-free) set S in a finite (abelian) group of odd order satisfies…”

Product-free subsets of groups, then and now

“…1 In some sources, one does require a = b. For instance, as noted in [9], I mistakenly assumed this in [11,Theorem 3].…”

Groups containing locally maximal product-free sets of size 4

“…Among other results, they calculated locally maximal product-free sets in groups of small orders, up to 16 in [11,12] as well as a few higher sizes. Giudici and Hart [9] started the classification of finite groups containing small locally maximal product-free sets. They classified finite groups containing locally maximal product-free sets of sizes 1 and 2, as well as some of size 3.…”

“…where SS = {xy | x, y ∈ S}, S −1 S = {x −1 y | x, y ∈ S}, SS −1 = {xy −1 | x, y ∈ S} and √ S = {x ∈ G | x 2 ∈ S} (see [9,Lemma 3.1]). Each (locally maximal product-free) set S in a finite group of odd order satisfies | √ S| = |S|.…”