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

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).