On the set of same-size conjugate classes

Abstract

For a finite group G, we define an equivalence relation ∼ on G as follows:$$\forall x,y\in G\hspace{1em}x\sim y\iff \left|x^G\right|=\left|y^G\right|,$$

where x^G is the conjugacy class of x in G. The set of sizes of equivalence classes with respect to this relation is called the same-size conjugate set of G and denote it by U(G). In this paper, we consider the influence of U(G) on the structure of a group G. In fact, we conjecture that all finite non-abelian simple groups G are characterizable by U(G) and we confirm this conjecture for the projective special linear groups PSL₃(3) and PSL₂(q), where $q\in \{5,7,8,9,17\}$.

Keywords:

• Conjugacy class
• simple group
• solvable group

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 20D99
• 20E45

Acknowledgments

I would like to thank the referees for their useful comments.

Additional information

Funding

This research was supported by the University of Kurdistan.