### 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 $a \notin \langle S \setminus \{a\}\rangle$.