具有几乎链条件的有限p-群
Finite p-groups with an almost chain condition
白鹏飞 1郭秀云 2王俊新1
作者信息
- 1. 山西财经大学应用数学学院,太原 030006
- 2. 上海大学数学系,上海 200444
- 折叠
摘要
设t为正整数.若一个有限p-群的所有指数为pt的子群皆交换,且它至少有一个指数为pt-1的非交换子群,则称它为At-群.若一个At-群恰有s个指数为pt-1的交换子群,其中s≥0,则称它为Ast-群.显然,对于任意有限非交换p-群,一定可找到合适的整数s和t使得它是一个Ast-群.为了深入研究有限非交换p-群,结合Ast-群的结构特点,本文描述具有A0t-1-子群的At-群的结构,并证明对于任意的Ast-群,若s≥1,则s≡1(mod p),进一步地,完全分类所有的A1t-群.
Abstract
For a positive integer t,a finite p-group is called an At-group if all its subgroups of index pt are abelian,but it has at least a non-abelian subgroup of index pt-1.An At-group is said to be an Ast-group if it has exactly s abelian subgroups of index pt-1,where s ≥ 0.For any finite non-abelian p-group,there must be suitable integers s and t such that it is an Ast-group.In order to study finite non-abelian p-groups deeply,combining with the structural characteristics of Ast-groups,we describe the structure of an At-group having an A0t-1-subgroup.We prove that for any Ast-group,if s≥1,then s ≡ 1(mod p),and completely classify the A1t-groups.
关键词
有限p-群/At-群/Ast-群/链条件Key words
finite p-group/At-group/Ast-group/chain condition引用本文复制引用
基金项目
国家自然科学基金(12171302)
国家自然科学基金(11801334)
山西省自然科学基金(202103021224287)
山西省高等学校科技创新项目(2021L278)
出版年
2024
NETL
NSTL
万方数据