A New Characterization Of Some Finite Simple Groups

We know class equation has a very great influence on the structure of finite groups in the finite groups theory. We can get the size of same order elements if we put some different conjugacy classes with the same order elements together, then we will get order equation of a finite group. In this paper, We give a new characterization of some simple K₃- groups except U₄(2) by their order equations and get the following result:Theorem 2.1 If G is a group, G (?) M if and only if G and M have the same orderequation, where M(?)A₅(2²Â·3·5),A₆(2³Â·3²Â·5),L₂(7)(2³Â·3·7),L₂(8)(2³Â·3²Â·7),L₂(17)(2⁴·3²Â·17),L₃(3)(2⁴·3³Â·13)or U₃(3)(2⁵·3³Â·7)．Generally, there is an intimate relation between the groups and graphs, and in many occasions properties of graphs can give rise to some properties of groups and vice versa. The author of this paper considers the alternating group A₁₀ and Lie type simple group L₂ (q) and prove the following theorems:Theorem 3.1 Suppose G is a finite group withâ–½(G)(?)â–½(A₁₀) then G(?)A₁₀,whereA₁₀ is the alternating group of degree 10.Theorem 3.2 Suppose G is a finite group withâ–½(G)(?)â–½(L₂(q)) then G(?)L₂(q).In 1987, Professor J.G. Thompson put forward the following conjecture in his letter to Professor W.J. Shi:Thompson' s Conjecture If G is a finite centerless group and M is a non-abelian simple group such that N(G) = N(M), then G (?) M, where N(G) := {n∈N | G has a conjugacy class of size n}.Nobody can completely solve Thompson' s conjecture, even give a counter example till now. It shows the difficulties in solving it.In this paper, we prove that Thompson' s conjecture is correct for U₄(2).Theorem 4.1 Suppose G, M are finite groups. If Z(G) = 1, N(G) = N(M), then G (?) M, where M = U)4(2).