Groups with the same number of centralizers

摘要

For any group G, let nacent(G) denote the set of all non-abelian centralizers of G. Jafarian Amiri and Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen87(3-4) (2015) 429-437 put forward the following question: Let H and G be finite simple groups. Is it true that if |nacent(H)| = |nacent(G)|, then G is isomorphic to H? In this paper, among other things, we give a negative answer to this question.