On finite CG-groups

Abstract

A finite group $G$ is a CG-group if $|\mathrm{Cent}(G)|=|G^{\prime }|+2$, where $G^{\prime }$ 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.