Sylow－正规化子属于群系F的有限群  

Finite Groups Whose Sylow-Normalizers Belong to a Formation F

作　　者：陈重穆[1] 

机构地区：[1]西南师范大学数学系

出　　处：《数学学报（中文版）》1996年第4期456-459,共4页Acta Mathematica Sinica：Chinese Series

摘　　要：设Ｆ是可解的，子群闭的，由｛ｆ（Ｐ）｝所局部定义的群系，Ｆｐ是由｛ｆ（ｑ）｝定义的ｐ－局部定义群系．Ｎ为幂零群系．本文证明了：１）设Ｆ满足：任一群属于Ｆ，当且仅当，对每ｐ．其ｐ－Ｓｙｌｏｗ－正规化子属于Ｆｐ．于是“群Ｇ∈Ｎ．Ｆ（幂零由Ｆ的扩张）的充要条件是，对每Ｐ，其ｐ－Ｓｙｌｏｗ－正规化子的Ｆｐ剩余次正规于Ｇ内．２）群Ｇ为超可解的充要条件是，对每ｐ，其ｐ－Ｓｙｌｏｗ－正规化子为ｐ－超可解，且其幂零剩余次正规于Ｇ内．若对每ｐ，群Ｇ的ｐ－Ｓｙｌｏｗ子群无商群与ｐ２－次对称群的ｐ－Ｓｙｌｏｗ子群同构，则称Ｇ为Ｂ－群．３）设Ｇ为Ｂ－群，又群系Ｆ含于σ－Ｓｙｌｏｗ塔群系内．于是①Ｇ∈Ｆ，当且仅当，对每ｐ，Ｇ的ｐ－Ｓｙｌｏｗ－正规化属于Ｆｐ；②Ｇ∈Ｎ·Ｆ，当且仅当，对每ｐ，Ｇ的ｐ－Ｓｙｌｏｗ－正规化子的Ｆｐ剩余在Ｇ内次正规．Suppose that F is a solvable and subgroups closed formation locally defined by{f(p)}. Fp is the formation locally defined byThe principal results of this paper as follow:1) If F holds that a group belongs to f when and only when for every prime p whosep-Sylow-normalizers belong in Fp, then a group G ∈ N. F (extentions of a nilpotent group bya F-group) when and only when for every prime p the Fp-residues of p-Sylow-normalizers aresubnormal in G.2) A group G is supersolvable when and only when for every prime p the p-Sylow-normalizersare supersolvable and whose nilpotent residues are subnormal in G.

关 键 词：正规化子 超可解群 有限群 SYLOW子群 

分 类 号：O152.1[理学—数学]