A Charactetization Of Finite Simple Groups

Let G be a finite group, M(G) the set of maximal Abelian subgroup of G, i.e M(G) = {|N|Nbe a Abelian subgroup of G, and for M < G,M Abelian, if N < M, then G = M or N = M). Especially, if G is Abelian, we may say M{G) = {1}.In this paper, we will prove the following results:Theorem 2.1 Let G be a finite group, then G A11 if and only if M(G) = M(A11).Theorem 2.2 Let G be a finite group, then G L2(q) if and only if M(G) = M(L2{q)), where q = 3,5(mod8), q = pnTheorem 2.3 Let G be a finite group, then G M if and only if M(G) = M(M), where M are sporadic simple groups except for Mathieu groups and Janko groups.Theorem 2.4 Let G be a finite group, then G G2(q) if and only if M(G) = M(2G2{q)), where q = 32m+1.Theorem 3.1 Let G be a finite group, then G D4(q) if and only if M(G) = M(3D4(q)), where q < 10.