On the conjecture of groups with the same number of centralizer

Abstract

For any group G, let $cent(G)$ denote the set of all centralizers of $G$. The authors in \cite{KZ}, Groups with the same number of centralizers, J. Algebra Appl. (2021) 2150012 (6 pages), posed the following conjecture: Let $G$ and $S$ be finite groups. Is it true that if $|Cent(G)|=|Cent(S)|$ and $|G^{\prime }|=|S^{\prime }|$, then $G$ is isoclonic to $S$? In this paper, among other things, we give a negative answer to this conjecture.