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 PSylp(M). Also we prove that if G is a finite group, then GM if and only if for every prime q, |NM(Q)|=|NG(Q)|, where QSylq(M) and QSylq(G).