A note on minimal subgroups of finite groups
A note on minimal subgroups of finite groups
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DOI：
10.1080/00927879608542654
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发表时间：
1996
影响因子：
0.7
通讯作者：
M. Asaad;A. Ballester-Bolinches;M. C. P. Aguilera
中科院分区：
文献类型：
--
作者：
M. Asaad;A. Ballester-Bolinches;M. C. P. Aguilera
本说明中考虑的所有小组都是有限的。回忆一下，有限群的极小子群是素数阶的子群。许多作者研究了有限群G的结构，假设G的所有最小子群都在群中。Ito [7;III， 5.3]证明了如果G是奇阶群，且G的所有极小子群都在G的中心，则G是幂零的。伊藤结果的一个推广是如下表述[7；IV,p。[35]：如果对于奇素数p， G的所有p阶子群都在G的中心，则G是p幂零的。如果G的所有2阶和4阶元素都在G的中心，那么G是2幂零的。Buckley[4]证明了如果G是奇阶群，且G的所有最小子群在G中都是正规的，则G是超溶的。后来Shaalan[8]证明了如果G是一个有限群，并且G的每一阶或4阶的子群在G中都是π-拟正规的，则G是超可溶的。回想一下，群G的子群H是…
1 Introduction and Preliminaries All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Ito [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Ito's result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is...