A note on abelian partitionable groups

Abstract

A group G has an abelian partition if it has a set theoretic partition into disjoint commutative subsets $A_0,A_1,\dots ,A_n$ where $\left|A_i\right|>1$ for all i and the identity is in A₀, such a group is called an abelian partitionable group ($AP-$group). The problem of classifying $AP-$groups was recently taken up by Mahmoudifar et al. who classified all groups with n = 2, 3. The motivation for this problem can be found in graph theory where partitions of graphs into induced complete subgraphs is of great importance. We achieve a partial classification of $AP-$groups and introduce a family of groups with no abelian partition.

Communicated by Mark L. Lewis

Keywords:

• Covering of groups
• dihedral group
• partition of groups
• self-centralizing involutions

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D06
• 05C25

Acknowledgments

The authors are grateful to the referee for the valuable comments and suggestions for the improvement of the paper. And the authors are indebted to Alireza Moghaddamfar for providing insightful comments which has resulted in improvements to this paper.