Counting centralizers and z-classes of some F-groups
Sekhar Jyoti Baishya

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 is called an F-group if for every , implies that . On the otherhand, two elements of a group are said to be -equivalent or in the same -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/ -classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of -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 -classes of some finite groups and extend some previous results.
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · graph theory and CDMA systems
