有限可解群的若干特征性质

SOME CHARACTERISTIC PROPERTIES OF SOLVABLE GROUPS

本文从极大子群和主因子的角度来讨论有限可解群，得到了有限可解群的若干新刻划。

In this paper,we explore the finite solvable groups by menas of maximal subgroups and chief factors of a finite group. We prove mainly the result: The following propositions are equivalent:(1) G is a solvable group;(2) For every maximal subgroup M of G,there exists a solvable normal subgroup H of G such that M∩H <H and M∩HG;(3) For every maximal subgroup M of G, there exists a normal subgroup H of G such that M ∩H<H, M∩ HG, and H/M∩H is solvable;(4) Every maximal subgroup M of G is complementary to an abelian central chief factor of G;(5) There is a nontrivial normal subgroup 11 of G such that 1 =H0<H1<..<Hn.= H is a section of a principal series of G,in which,G/CG(Hi+1/Hi)is solvable,0≤i<n,and(6) For any minimal normal subgroup N/Φ(G) of G/Φ(G),G/Cc(N/Φ(G) ) is solvable;(7) For any Abelian minimal normal subgroup H/Φ(G) of G/Φ(G),G/CG(N/Φ(G))and CG(F(G)/ Φ(G))are solvable.