Groups that have a partition by commuting subsets

Tuval Foguel 1 , Josh Hiller 2 , Mark L. Lewis 3 , and Alireza Moghaddamfar 4
  • 1 Department of Mathematics and Computer Science, Adelphi University, 11010, Garden City, NY, USA
  • 2 Department of Mathematics and Computer Science, Adelphi University, NY 11010, Garden City, USA
  • 3 Department of Mathematical Sciences, Kent State University, Ohio 44242, Kent, USA
  • 4 K. N. Toosi University of Technology, Faculty of Mathematics, Tehran, Iran
Tuval Foguel, Josh Hiller, Mark L. Lewis and Alireza Moghaddamfar

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 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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