Pure Quantitative Characterization Of Finite Simple Groups

Groups considered in the thesis are finite groups and simple groups are non-Abelian.Let G be a finite group. Denote by e(G) the set of all orders of elements in G, by h( e(G)) the number of isomorphism classes of groups H such that e(H) = e(G), and by the order of G.A group G is said to be characterizable only by its set of element orders (recognizable, irrecognizable respectively) if h(e(G)) = 1 ( h(e(G)) ,h(e(G)) = repectively).In 1989, Professor Shi Wujie put forward the following conjecture:Conjecture Let G be a group and M a finite simple group. Then G M if and only if (1) e(G) = e(M); (2) = The author discusses the above conjecture and proves the following Theorem A and Theorem B in Section 2 and Section 3:Theorem A Let G be a group and M(q) a Lie type simple group 2Dn(q), n 4 or Dt(q),l 5, l odd. Then G M(q) if and only if (1) e(G) = e(M(q)); (2) = Theorem B Let G be a group and a finite Symplectic simple group. Then if and only if (1) Though the above conjecture characterizes some finite groups by two conditions, many finite groups are characterized by only the set of their element orders. The author obtains the following Theorem C in Section 4:Theorem C Let G be a group and L = L3(3(2m-1)),m 2, or G2(3n). Then G L if and only if e(G) = e(L).Note that h(e(L3(3))) = oo. Professor Shi Wujie and Professor V.D.Mazurov give four pairs of groups G such that h(e(G)) = 2:(1) L3(5),L3(5).2; (2) L3(9),L3(9).21, (3) S6(2),O8+(2); (4) O7(3),O8+(3).In the last section the author discusses an open probelem, which was put forward by Japenese Professor S.Abe and Professor N.Iiyori in 2002, and proves the following Theorem D.Probelem(Abe-Iiyori) Let G be a finite group and S be an non-abelian simple group. Assume that the set ord(Ssol(G)) of orders of solvable subgroups of G coincides with ord(Ssol(S)). Then is G isomorphic to 5?Theorem D Let G be a finite group and S be one of spordic simple groups. Then G S if and only if the set ord(Ssol(G)) coincides with ord(Ssol[(S)).