Groups that have a partition by commuting subsets

Abstract

Abstract Let 𝐺 be a nonabelian group. We say that 𝐺 has an abelian partition if there exists a partition of 𝐺 into commuting subsets A 1 , A 2 , … , A n A_{1},A_{2},\ldots,A_{n} of 𝐺 such that | A i | ⩾ 2 \lvert A_{i}\rvert\geqslant 2 for each i = 1 , 2 , … , n i=1,2,\ldots,n . This paper investigates problems relating to groups with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions – with exact calculations in some cases. Finally, we examine how the size of a partition with the minimum number of parts behaves with respect to the direct product.