Study Of Several Problems About The Quantitative Property Of Finite Groups

Let G be finite groups. Denote k(G) to be the number of conjugacy classes of G, and Ï€_e(G) to be the set of orders of the elements of G. Then there exists nonnegtive number k such that, k(G) = |Ï€_e(G)| + k. We call this groups to be co(k) groups.Syskin puts forward a famous cinjecture in 1980 that: Let G be finite groups, if any same order elements of G are conjugate, then G ≌ 1, Z₂, S₃.P. Fizpatric, W. Feit and J. P. Zhang have solved this well-known conjecture during 1985 -1988 independently. When k = 0, the groups difined above become co(0) groups, they are just the groups in Syskin conjecture.ALso Let M_i(G) = {x ∈ G|o(x) = i, i ∈ Ï€_e(G)}, particularly, let M(G) = M_k(G), where k = maxÏ€_e(G). Thompson once gave a conjecture that: Let G₁, G₂ be finite groups, if |M_i(G₁)| = |M_i(G₂)|(i ∈ Ï€_e(G)), then if G₁ is solvable, then G₂ is solvable too.This paper investigates the following problems:(1) Finite co(k) groups satisfying some conditions.(2) Classify all finite co(l) groups.(3) Classify all finite solvable co(2) groups.(4) Finite groups with exactly 4p or 4p² maximal order elements. The main results of this paper are following five theorems:Theorem A Let G be finite co(k) groups, N be solvable normall subgroup of G, then G/N is co(i) group, 0 ≤ i ≤ k.Theorem B G is co(l) group if and only if G isomorphic to one of the following groups:A₅, L₂{7), S₅, S₄, A₄, Hol(Z₅), Z₃ : Z₄, D₁0, Z₃, Z₄.Theorem C let G be finite solvable group, then G is co(2) group if and only if G isomorphic to one of the following groups:Z2 x Z2, Z6, D8, QB, Du, Z7 : Z3, Z7 : Z6, Z15 : Z4, S3xZ2, D18,QS : Z3, (Z3 x Z3) :Theorem D let G be a finite group, if G has exactly Ap maximal order elements, where p is a prime, then G is solvable, unless G = S$.Theorem E Let G be a finite group, if G has exactly 4p2 maximal order elements, where p is a prime, then G is solvable.