2016/11/20 by Chimere Anabanti, Anabanti, Chimere · 1 citation
Mathematics · #05E15 #20K01 #Combinatorics (math.CO) #FOS: Mathematics #Primary 11B75 #Secondary 20D60 #math.CO #msc:05E15 #msc:11B75 #msc:20D60 #msc:20K01
paper · pdf · doi:10.48550/arxiv.1611.06546
arxiv created 2016/11/20 · arxiv updated 2016/11/22
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 2n-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 3n-1. Finally, for prime p>3 and n∈ ℕ, we show that there is no direct analogue of this result for elementary abelian p-groups of finite rank n.