On 10-Centralizer Groups of Odd Order

Z Foruzanfar,Z Mostaghim

doi:10.1155/2014/607984

Abstract

Let G be a group, and let Cent(G) denote the number of distinct centralizers of its elements. A group G is called n-centralizer if Cent(G)=n. In this paper, we investigate the structure of finite groups of odd order with Cent(G)=10 and prove that there is no finite nonabelian group of odd order with Cent(G)=10.

Highlights

Throughout this paper all groups mentioned are assumed to be finite, and we will use usual notation; for example, Cn denotes the cyclic group of order n, and Cn ⋊ Cp denotes the semidirect product of Cn and Cp with normal subgroup Cn, where n is a positive integer and p is a prime
Many authors have studied the influence of |Cent(G)| on finite group G
It is clear that a group G is 1-centralizer if and only if it is abelian

Summary

Introduction

Throughout this paper all groups mentioned are assumed to be finite, and we will use usual notation; for example, Cn denotes the cyclic group of order n, and Cn ⋊ Cp denotes the semidirect product of Cn and Cp with normal subgroup Cn, where n is a positive integer and p is a prime. For a group G, Z(G) denotes the center of G, and Cent(G) = {CG(x) | x ∈ G}, where CG(x) is the centralizer of the element x in G; that is, CG(x) = {y ∈ G | xy = yx}. There is no finite nonabelian group G of odd order with |Cent(G)| = 10.

Results

Conclusion