Journal of Group Theory ( IF 0.466 ) Pub Date : 2020-11-04 , DOI: 10.1515/jgth-2020-0065
Tuval Foguel; Josh Hiller; Mark L. Lewis; Alireza Moghaddamfar

Let 𝐺 be a nonabelian group. We say that 𝐺 has an abelian partition if there exists a partition of 𝐺 into commuting subsets $A1,A2,…,An$ of 𝐺 such that $|Ai|⩾2$ for each $i=1,2,…,n$. 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.

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