# groups with the same number of centralizers

## TL;DR

This paper investigates the relationship between the number of nonabelian centralizers in finite simple groups and their isomorphism classes, providing a counterexample to a previously posed conjecture.

## Contribution

It demonstrates that having the same number of nonabelian centralizers does not necessarily imply isomorphism between finite simple groups.

## Key findings

- Counterexample to the conjecture by Amiri and Rostami
- Shows non-uniqueness of nonabelian centralizer counts in finite simple groups
- Challenges assumptions about group invariants and isomorphism

## Abstract

For any group $G$, let $nacent(G)$ denote the set of all nonabelian centralizers of $G$. Amiri and Rostami in (Publ. Math. Debrecen 87/3-4 (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.