Groups that have a Partition by Commuting Subsets

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 $A_1,A_2,\dots ,A_n$ of $G$, such that $|A_i|⩾2$ for each $i=1,2,\dots ,n$. This paper investigates problems relating to group 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 partitions with the minimum number of parts behaves with respect to the direct product.

19 pages - revised according to referee's report