  Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical
Society, 2018] recently gave a sharp count on the number of maximal sum-free
subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and
Erd\H{o}s. In contrast, not as much is know about the analogous problem for
finite abelian groups. In this paper we give the first sharp results in this
direction, determining asymptotically the number of maximal sum-free sets in
both the binary and ternary spaces $\mathbb Z^k_2$ and $\mathbb Z^k_3$. We also
make progress on a conjecture of Balogh, Liu, Sharifzadeh and Treglown
concerning a general lower bound on the number of maximal sum-free sets in
abelian groups of a fixed order. Indeed, we verify the conjecture for all
finite abelian groups with a cyclic component of size at least 3084. Other
related results and open problems are also presented.
