International Journal of Mathematics and Mathematical Sciences 
Volume 2005 (2005), Issue 9, Pages 1449-1453
doi:10.1155/IJMMS.2005.1449

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

Behrooz Khosravi1,2 and Behnam Khosravi3

 1Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, Tehran 15914, Iran
2Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran
3Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin, Tehran 19838, Iran
Received 17 October 2004

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).