子群为拟正规或自正规的有限群

Finite Groups with Only Quasinormal and Selfnormal Subgroups

本文研究了每个子群为拟正规或自正规的有限群，给出了这类群的完全分类，主要结果为定理Ｇ的每个子群为拟正规或自正规当且仅当Ｇ为下列情形之一：Ⅰ）Ｇ为拟Ｈａｍｉｌｔｏｎ群，Ⅱ）Ｇ＝ＨＰ，其中Ｈ为Ｇ的正规ａｂｅｌｉａｎｐ＇－Ｈａｌｌ子群．Ｐ＝〈ｘ〉∈Ｓｙｌ＿ｐ（Ｇ）。〈ｘ￣ｐ〉＝Ｏ＿ｐ（Ｇ）＝Ｚ（Ｇ），ｘ在Ｈ上诱导Ｈ的一个ｐ阶无不动点的幂自同构．ｐ为｜Ｇ｜的最小素因子。由此定理可得文［１］所获得的定理。

In this paper authors study such a finite group in which each subgroup is eitherquasinormal or selfuormal and characterize this kind of finite groups. The main result is Theorem Every subgroup of group G is either quasinormal or selfuormal if and only ifG is precisely one of the followingⅠ)G is a quasi-Hamilton group.Ⅱ)G=HP, where H is a normal abelian p'-Hall subgrou.P=〈x〉∈Syl_p(G),〈x￣p〉=O_p(G)=Z(G)and x induces a fixed-point-free power automorphisim of order p on H. p is thesmallest prime factor of |G|.The theorem in paper[1]can be obtained by the above theorem.