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Groups that have a partition by commuting subsets

  • Tuval Foguel , Josh Hiller , Mark L. Lewis EMAIL logo and Alireza Moghaddamfar
Published/Copyright: November 4, 2020
Journal of Group Theory
From the journal Journal of Group Theory Volume 24 Issue 3

Abstract

Let ๐บ be a nonabelian group. We say that ๐บ has an abelian partition if there exists a partition of ๐บ into commuting subsets A1,A2,โ€ฆ,An of ๐บ such that |Ai|โฉพ2 for each i=1,2,โ€ฆ,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 โ€“ 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.

In memory of Professor Carlo Casolo

  1. Communicated by: Andrea Lucchini

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Received: 2020-05-01
Revised: 2020-10-06
Published Online: 2020-11-04
Published in Print: 2021-05-01

ยฉ 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The groups (2, ๐‘š | ๐‘›, ๐‘˜ | 1, ๐‘ž): Finiteness and homotopy
  3. Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem
  4. Metabelian groups: Full-rank presentations, randomness and Diophantine problems
  5. Nilpotence relations in products of groups
  6. The subnormal structure of classical-like groups over commutative rings
  7. A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups
  8. On the stabilisers of points in groups with micro-supported actions
  9. Groups that have a partition by commuting subsets
  10. Commuting graph of ๐ด-orbits
  11. On the holomorph of finite semisimple groups
  12. TI subgroups and depth 3 subgroups in simple Suzuki groups
  13. On the minimal degree of a transitive permutation group with stabilizer a 2-group
  14. Exponent of a finite group admitting a coprime automorphism of prime order
https://doi.org/10.1515/jgth-2020-0065

Articles in the same Issue

  1. Frontmatter
  2. The groups (2, ๐‘š | ๐‘›, ๐‘˜ | 1, ๐‘ž): Finiteness and homotopy
  3. Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem
  4. Metabelian groups: Full-rank presentations, randomness and Diophantine problems
  5. Nilpotence relations in products of groups
  6. The subnormal structure of classical-like groups over commutative rings
  7. A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups
  8. On the stabilisers of points in groups with micro-supported actions
  9. Groups that have a partition by commuting subsets
  10. Commuting graph of ๐ด-orbits
  11. On the holomorph of finite semisimple groups
  12. TI subgroups and depth 3 subgroups in simple Suzuki groups
  13. On the minimal degree of a transitive permutation group with stabilizer a 2-group
  14. Exponent of a finite group admitting a coprime automorphism of prime order
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