On the conjecture of groups with the same number of centralizers.

For any group G, let cent(G) denote the set of all centralizers of G. K. Khoramshahi and M. Zarrin [Groups with the same number of centralizers, <italic>J. Algebra Appl.</italic> 20 (2021) 2150012], posed the following conjecture: Let G and S be finite groups. Is it true that if |Cent(G)| = |Cent(S)| and |G′| = |S′|, then G is isoclonic to S? In this paper, among other things, we give a negative answer to this conjecture. [ABSTRACT FROM AUTHOR]