On the supersolvability of finite groups. I
On the supersolvability of finite groups. I
复制标题
DOI：
10.1007/bf01917519
复制
发表时间：
1981-03
期刊：
影响因子：
--
通讯作者：
M. Asaad
中科院分区：
文献类型：
--
作者：
M. Asaad
J.Buckley[1]证明了：如果G是奇阶有限群，且每个极小子群都是正规的，则G是超可解的。在[2]中，Robert W.van der WAALL证明了这一结果是DOERK一个定理的直接推论；见[3]。如果对G中的x，H与(H，Hx)中的Hx共轭，则G的子群H称为G中的原正规群。已知G的子群H在G中正规当且仅当它在G中次正规和亚正规(练习6，文献[4]第14页)。这一点和Doerk定理的一个直接结果是Buckley结果的如下推广。
It was proved by J. BUCKLEY [1] that if G is a finite group of odd order in which every minimal subgroup is normal then G is supersolvable. In [2], ROBERT W. VAN DER WAALL showed that this result is a direct corollary of a theorem of DOERK; see [3]. A subgroup H of G is called pronormal in G if for x in G, H is conjugate to H x in (H, HX). It is known that a subgroup H of G is normal in G iff it is both subnormal and pronormal in G (Exercise 6, p. 14 of [4]). An immediate consequence of this and Doerk's Theorem is the following generalization of Buckley's result.