Characterization Of Finite Simple Groups By The Number Of Singular Elements

The quantitative characterization of finite groups is an important research subject.Finite simple groups are bricks of finite groups.In this thesis,we study the characterization of some nonabelian simple groups by use of the set of numbers of singular elements.Let G be a finite group,and π(G)the set of all primes of |G|.Set r∈π(G),we called a r-singular element of G if its order is divisible by r.Denote by μr(G)the probability of r-singular elements in G,Let μ(G):={μr(G)|r∈π(G)}.In this thesis,we characterize some linear simple groups and alternating simple groups by μ(G).Moreover,we obtain following main results:Theorem Let G be a finite group,and S the following simple groups:(1)S=PSL(2,p),where p is an odd prime number;(2)S=PSL(2,2m),where m≥2;(3)S=An,where 5≤n≤18,n≠10.If μ(G)=μ(S),then G(?)S.