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.

On solubility of groups with finitely many centralizers

“…This fact is a correction of the proof of Theorem B (2) in [12]. Moreover, we extend the Theorem 3.5 in [8] in the primitive case.…”

“…Using similar arguments we extend Theorem 3.5 in [8] and for the case n = 11, we add the hypothesis that H is a primitive 11-centralizer group.…”