幂零Hall π-子群的存在性

The Existence of Nilpotent Hall π Subgroups

本文首先将Ｈａｌ定理推广为：设Ｎ为Ｇ的正规子群，若Ｎ为Ｅｎπ群，Ｇ／Ｎ为Ｄπ群，则Ｇ为Ｄπ群．在此基础上得到了群Ｇ为Ｅｎπ群的充要条件为：（１）Ｇ存在正规子群Ｎ，满足Ｎ及Ｇ／Ｎ为Ｅｎπ群；（２）对任意ｐ∈π，任意ｑ∈π｛ｐ｝及任意ｐ元素ｘ，ＣＧ（ｘ）含Ｇ的Ｓｙｌｏｗｑ子群．另外，我们对非Ａｂｌｅ单群的情形也进行了一些讨论．

In this paper the Hall's theorem is first generalized as follows: Let N be a normal subgroup of a finite group G . If N is an E n π group and G/N is a D π group, then G is a D π group. On the basis of this, we show that a finite group G is an E n π group if and only if (1) there exists a normal subgroup N of G such that N and G/N are E n π groups; and (2) for any p∈π, any q∈π-{p} , and any p element x,C G(x) contains a Sylow q subgroup of G . At the same time we have some discussions for nonabelian simple groups.