Groups with the same number of centralizers

Abstract

For any group [Formula: see text], let [Formula: see text] denote the set of all non-abelian centralizers of [Formula: see text]. Jafarian Amiri and Rostami, Groups with a few nonabelian centralizers, Publ. Math. Debrecen 87(3–4) (2015) 429–437 put forward the following question: Let [Formula: see text] and [Formula: see text] be finite simple groups. Is it true that if [Formula: see text], then [Formula: see text] is isomorphic to [Formula: see text]? In this paper, among other things, we give a negative answer to this question.