Groups with the same orders of Sylow normalizers as the Mathieu groups

Behrooz Khosravi,Behnam Khosravi

doi:10.1155/ijmms.2005.1449

Abstract

There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. Let M be a Mathieu group and let p be the greatest prime divisor of |M|. In this paper, we prove that M is uniquely determined by |M| and |NM(P)|, where P ∈ Sylp(M). Also we prove that if G is a finite group, then G≅M if and only if for every prime q, |NM(Q)| = |NG(Q′)|, where Q ∈ Sylq(M) and Q′ ∈ Sylq(G).

Highlights

IntroductionWe denote by π(G) the set of all prime divisors of |G|
It was proved that if G is an alternating group, a finite projective special linear group, a Janko sporadic simple group, or a finite projective special symplectic group, G is characterizable by the orders of normalizers of its Sylow subgroups [1, 2, 3, 4, 10]
Mazurov and Shi [11, 12, 13, 14] and Deng [7] proved that some of the almost sporadic simple groups are characterizable by the set of element orders

Summary

Introduction

We denote by π(G) the set of all prime divisors of |G|. It was proved that if G is an alternating group, a finite projective special linear group, a Janko sporadic simple group, or a finite projective special symplectic group, G is characterizable by the orders of normalizers of its Sylow subgroups [1, 2, 3, 4, 10]. Mazurov and Shi [11, 12, 13, 14] and Deng [7] proved that some of the almost sporadic simple groups are characterizable by the set of element orders. Khosravi [9] proved that some of the almost sporadic simple groups are characterizable by the set of order components. Z6 ∼= S3 and they are not characterizable by these conditions

Preliminary results

Characterizations of the Mathieu groups