摘要
从极大子群、中心主因子和正规子群的G 主列的角度来讨论有限幂零群,获得了有限幂零群的若干新刻划.设M是有限群G的任一极大子群,H G.令 G=G/Φ(G),则G是幂零群当且仅当下列条件之一成立:(1)如H≤\M,则H∩M G且H/H∩M≤Z(G/H∩M);(2)如H≤\M,则M≤CG(H/H∩M);(3)如H≤\M,则H≤CG(M/H∩M);(4)如H≤\M,则M补于G的一个中心主因子;(5)F( G)有一个 G 主列,其中每个主因子都是 G中心的且CG(F( G))可解;(6)Soc( G)有一个 G 主列,其中每个主因子都是 G 中心的;(7)K∞(G)≤H,H/Φ(H)有一个 G 主列其中每个主因子都是 G 中心的;(8)HCG(H)≤Z∞(G).
In this paper, we explore finite nilpotent groups by means of maximal subgroups, central chief factors and G-principal serieses of normal subgroups of finite groups. We obtain some new chracterizations of finite nilpotent groups. Let M be any maximal subgroup of a finite group G and H be a normal subgroup of G. Put =G/Φ(G). Then G is nilpotent if and only if one of the following conditions holds:(1) if H≤\M then H∩MG and H/H∩M≤Z(G/H∩M); (2) if H≤\M then M≤C_G(H/H∩M); (3) if H≤\M then H≤C_G(M/H∩M); (4) if H≤\M then M is complementary to a central chief factor of G; (5) F() has a -principal series whose every chief factor is -central and C_G(F()) is solvable; (6) Soc () has a -principal series whose every chief factor is -central; (7) K_∞(G)≤H and H/Φ(H) has a -principal series whose every chief factor is -central; (8) HC_G(H)≤Z_∞(G).
出处
《四川师范大学学报（自然科学版）》
CAS
CSCD
2004年第5期441-445,共5页
Journal of Sichuan Normal University（Natural Science）
基金
四川省学位委员会和四川省教育厅重点学科建设基金资助项目
关键词
有限幂零群
极大子群
中心主因子
正规子群的G-主列
Finite nilpotent group
Maximal subgroup
Central chief factor
G-principal series of a normal subgroup
