子群皆半正规的有限群

FINITE GROUPS WHOSE EVERY SUBGROUP IS SEMI-NORMAL

本文的主要目的是证明如下两个定理：Ⅰ．对于有限群Ｇ，下列命题等价：（１）Ｇ的ｓｙｌｏｗ子群皆半正规；（２）Ｇ的子群皆半正规；（３）Ｇ的子群皆Ｓ－半正规；（４）Ｇ的Ｓｙｌｏｗ子群皆强半正规；（５）Ｇ的子群皆半正规或自正规；（６）Ｇ的子群皆Ｓ－半正规或自正规；（７）Ｇ是广幂零群；令Ｈ／Ｋ是Ｇ的任一主因子，则Ｇ／ＣＧ（Ｈ／Ｋ）是阶与｜Ｈ／Ｋ｜互素的素数幂阶循环群．Ⅱ．对于有限群Ｇ，下列命题等价：（１）Ｇ的子群皆Ｓ－半正规或自正规，且Ｇ确有自正规的Ｓｙｌｏｗ子群；（２）Ｇ＝ＰＨ，Ｐ∩Ｈ＝１．其中Ｐ∈Ｓｙｌｐ，（Ｇ），Ｐ＝ｍｉｎ（π（Ｇ）），ＮＧ（Ｐ）＝Ｐ，Ｈ＝Ｋ∞（Ｇ）是Ｇ的幂零剩余，并且Ｋ≤Ｈ，均有Ｋ Ｇ；（２）Ｇ＝ＰＨ，其中Ｐ∈Ｓｙｌｐ，（Ｇ），ｐ＝ｍｉｎ（π（Ｇ）），Ｈ是Ａｂｅｌ群，并且Ｐ不能平凡作用于Ｈ的任一≠１的子群，Ｐ的每个≠１的元诱导Ｈ的一个幂自同构．

The main purpose of the present paper is proving the following two theorems: I. For a finite groupG, the following statements are equivalent: (1)every Sylow group of G is semi-normal; (2)every subgroup of G issemi-normal ; (3)every subgroup of G is S-semi-normal; (4) every Sylow subgroup of G is strong-semi-normal;(5)every subgroup of G is semi-normal or self-normal ; (6)every subgroup of G is S -semi-normal or self-normal;(7) G is generalized nilpotent ; let H/K be any chief factor of G, G/CG(H/K) is a cyclic group of order a primePower which is coprime with |H/K|. Ⅱ. For a finite group G, the following three statements are equivalent: (1)every subgroup of Cis S -semi-normal or self-normal, and C has truly self-normal Sylow subgroups; (2) G=PH,p ∩H = 1, in which, P ∈sylp(G), p = min(π(G)), NG(P) = P, H = K∞(G ), is the nilpotent residual of G,and KG, K ≤ H; (3) G= PH, in which P ∈Sylp(G), p = min(π(G )), H is abelian, and P can not acttrivially on any non-trivial subgroup of H, and every non-identity element of P induces a power automorphism of Hby conjugating.