Characterizations Of Some Finite Simple Groups

It is well known that finite simple groups are bricks of finite groups. In order to know better their properties and structures, it is very helpful for us to characterize them by their obvious, simple and instinctive properties.In this doctoral dissertation we will study the following questions:quasirecognition by prime graph of finite simple groups; characterization of finite simple groups by using group order and the maximal order of elements:characterization of finite simple groups by the set of the number of elements with the same order.This paper is divided into four chapters, the following is main contents:In Chapter 1, we introduce some usual symbols, basic definitions.In Chapter 2, we discuss quasirecognition by prime graph of finite simple groups, and get the following results:Theorem A G2(q) (q= 32n+1) is quasirecognizable by prime graph.Theorem B 2B2(q)(q = 22n+1> 2) is quasirecognizable by prime graph.Theorem C E7(q) (q= 2,3) is quasirecognizable by prime graph.Theorem D Let G be a finite group, then G(?) Sp if and only if|G|=|SP|andΓ(G)=Γ(Sp), where p is a prime.In Chapter 3, we discuss characterization of finite simple groups by using group order and the maximal order of elements, and get the following results:Theorem E let G be a finite group and H one of simple K3-groups. We denote by k(G) the maximal order of elements in G. Then the following cases hold:(1) If H (?) L2(7), U4(2), then|G|=|H|, k(G)= k(H) if and only if G= H.(2) If H(?) L2(7), then|G|=|L2(7)|, k(G)= k(L2(7)) if and only if G= L2(7) or G is a 2-Frobenius group, this moment, G(?) Z3[Z7[P]], where P(?)G, G/P(?)Z3[Z7], P is an elementary abelian 2-group of order 23 andπe(G)={1,2,3,6,7}, whereπe(G) is used to denote the set of orders of elements in G.(3) If H(?)U4(2), then|G|=|U4(2)|, k(G)= k(U4(2)), k(G)=k1(U4(2)) if and only if G(?)U4(2), where k1(G) is used to denote the second largest element order of G.Theorem F|G|=|L2(p)|,k(G)=k(L2(p)) if and only if G(?)L2(p), where p=8n±3>3 is a prime and n is a natural number.In Chapter 4, we discuss characterization of finite simple groups by the set of the number of elements with the same order, and get the following result:Theorem G L2(2n) (n=4,5,7) can be determined by the set of the number of elements with the same order.