Counting centralizers and z-classes of some F-groups

TL;DR

This paper characterizes conditions under which finite groups have equal numbers of centralizers and z-classes, especially focusing on F-groups, and computes these quantities for specific finite groups.

Contribution

It provides necessary and sufficient conditions for the number of centralizers and z-classes to match the index of the center in finite groups, extending previous results.

Findings

Conditions for equal number of centralizers and z-classes in finite groups.

Necessary and sufficient conditions for z-class count to be maximal.

Computed centralizers and z-classes for specific finite groups.

Abstract

A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their centralizers are conjugate in the group. In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of $z$-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and $z$-classes of some finite groups and extend some previous results.