  Tarnauceanu [Archiv der Mathematik, 102 (1), (2014), 11--14] gave a
characterisation of elementary abelian $2$-groups in terms of their maximal
sum-free sets. His theorem states that a finite group $G$ is an elementary
abelian $2$-group if and only if the set of maximal sum-free sets coincides
with the set of complements of the maximal subgroups. A corollary is that the
number of maximal sum-free sets in an elementary abelian $2$-group of finite
rank $n$ is $2^n-1$. Regretfully, we show here that the theorem is wrong. We
then prove a correct version of the theorem from which the desired corollary
can be deduced. Moreover, we give a characterisation of elementary abelian
$3$-groups in terms of their maximal sum-free sets. A corollary to our result
is that the number of maximal sum-free sets in an elementary abelian $3$-group
of finite rank $n$ is $3^n-1$. Finally, for prime $p>3$ and $n\in \mathbb{N}$,
we show that there is no direct analogue of this result for elementary abelian
$p$-groups of finite rank $n$.
