Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the conjecture of groups with the same number of centralizer

N. Ahmadkhah, M. Zarrin

Abstract

For any group G, let $cent(G)$ denote the set of all centralizers of $G$. The authors in \cite{KZ}, Groups with the same number of centralizers, J. Algebra Appl. (2021) 2150012 (6 pages), 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.

On the conjecture of groups with the same number of centralizer

Abstract

For any group G, let denote the set of all centralizers of . The authors in \cite{KZ}, Groups with the same number of centralizers, J. Algebra Appl. (2021) 2150012 (6 pages), posed the following conjecture: Let and be finite groups. Is it true that if and , then is isoclonic to ? In this paper, among other things, we give a negative answer to this conjecture.
Paper Structure (3 sections, 13 theorems, 2 equations)

This paper contains 3 sections, 13 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Characterization of $Cpo$-groups
  3. Groups with the Same number of centralizers

Key Result

Lemma 2.2

Let $G$ be a $Cpo$-group and $H$ be a non-centerless subgroup of $G$. Then $|H|$ is prime.

Theorems & Definitions (28)

  • Conjecture 1.1
  • Definition 1.2
  • Example 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Corollary 2.6
  • ...and 18 more