A characterisation of elementary abelian 3-groups

TL;DR

This paper corrects a previous characterization of elementary abelian 2-groups using sum-free sets and extends the analysis to elementary abelian 3-groups, providing exact counts of maximal sum-free sets and discussing limitations for primes greater than 3.

Contribution

It corrects a flawed theorem about elementary abelian 2-groups and establishes a new characterization for elementary abelian 3-groups based on their maximal sum-free sets.

Findings

Corrected the characterization of elementary abelian 2-groups.

Derived the number of maximal sum-free sets in elementary abelian 3-groups as 3^n - 1.

Showed no similar characterization exists for primes p > 3.

Abstract

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 $n$.