A characterization of elementary abelian 2-groups (Q2447663)

For an additive (but not necessarily abelian) group \(G\), a subset \(A\) is called \textit{sum-free} [\textit{I. Schur}, Jahresber. Dtsch. Math.-Ver. 25, 114--117 (1916; JFM 46.0193.02)] if \((A+A)\cap A=\varnothing \). The main result is: Theorem 1.1. A finite group \(G\) is an elementary Abelian \(2\)-group if and only if the set of maximal sum-free subsets coincides with the set of complements of maximal subgroups. Using this, some numbers are computed: Corollary 1.2. The number of maximal sum-free subsets of \(\mathbb{Z}_2^n\) is \(2^n-1\), and each of them has \(2^{n-1}\) elements.