The Influence By Elements Of Maximal Order In A Finite Group On The Structure Of The Finite Group

In 1987 the winner of Fields Prize J. G. Thompson announced two conjectures:Conjecture 1 Let G be a finite group, N(G) = {n|there exists a conjugacy class C such that |C| = n}. If Z(G) = 1, M is a non-abelian simple group, such that N(G) = N(M), then G(?)M.Conjecture 2 Let G and M are finite groups, such that |Ml(G)| = |Ml(M)| , for l = 1 , 2 , .... If G is solvable, then M is solvable.G. Y. Chen once discussed these two conjectures and proved that if the simple groups M has non-connected prime graphs, the then conjecture one holds in 1994. But for the second conjecture, it is still open without any progress. But some related results obtained in past years, which are about the influence on the number |M(G)| of elements of maximal order in a finite group G. Such topic was first discussed in [10]. And proved that if |M(G)| is 2 or an odd number or (?)(k), where k is the largest element order in G, then G is solvable. Afterwards it is proved that G is solvable if |M(G)| = 8 in [11], if |M(G)| ≤ 20 in [12], |M(G)| = 2p² in [13]. In [14], it is proved that G is solvable if Ï€_e(G) = {1,2,3,5,6}. And then for |M(G)| = 32, 2p³, 2m (m odd), it is proved that G is solvable too (see [15],[16] and [17]. All these discussions stop after proving G is solvable. In [3], the structure of finite groups with |M(G)| = 30 is given. Of course, all these discussions are helpful to the research on the conjecture 2.In this paper, we prove the following ten theorems:Theorem 1.1 suppose G is a finite group having 42 elements of maximal order, then G is one of following groups:(1) G (?) [Z₄₃]. H, where [Z₄₃] (?) G, H (?) Z₂ × Z₃ × Z₇.(2) G has a normal subgroup Z_k with order k (k = 49,86,98) and G/Z_k (?) Z₂ × Z₃ × Z₇.