有限群的极大且正规子群的交

The Intersection of the Maximum and Normal Subgroups of a Finite Group

设Ｇ是有限群，φ（Ｇ）是Ｇ的极大且正规子群的交。讨论了φ（Ｇ）的一些性质，并得到了一个正规π－补定理。设φ（Ｇ）是有限群Ｇ的极大且正规子群的交，则φ（Ｇ）是Ｇ的所有正规非生成元集合；设π是素数集，Ｈ是Ｇ的幂零Ｈａｌπ－子群。则Ｇ有正规π－补当且仅当Ｈ∩φ（Ｇ）＝Φ（Ｈ）。其中Φ（Ｈ）为Ｈ的Ｆｒａｔｉｎｉ子群。

Let G be a finite group and φ(G) the intersection of the maximum and normal subgroups of G .Some properties of φ(G) is discussed and a theorem of normal π complements is obtained. Theorem 1 φ(G) is the set of all normal non generators of G . Theorem 2 Let π be a set of primes and H a nilpotent Hall π subgroup.Then H has a normal π complement if and only if H∩φ(H)=Φ(H) ,where Φ(H) is the Frattini subgroup of H .