On finite CG-groups

TL;DR

This paper investigates the properties of finite CG-groups, providing new results and disproving a previous conjecture related to the structure of these groups.

Contribution

It introduces new findings on finite CG-groups and refutes an existing conjecture about their characteristics.

Findings

Established new properties of finite CG-groups

Disproved Khoramshahi and Zarrin's conjecture 2.3

Enhanced understanding of the relationship between centralizers and commutator subgroups

Abstract

A finite group $G$ is a CG-group if $|{\rm Cent}(G) | = | G' |+2$, where $G'$ is the commutator subgroup and Cent$(G)$ is the set of distinct element centralizers of $G$. In this paper we give some results on CG-groups. We also give a negative answer to \cite[Conjecture 2.3]{con} given by K. Khoramshahi and M. Zarrin.