摘要
设t为正整数.若一个有限p-群的所有指数为p^(t)的子群皆交换,且它至少有一个指数为p^(t−1)的非交换子群,则称它为A_(t)-群.若一个A_(t)-群恰有s个指数为p^(t−1)的交换子群,其中s>0,则称它为A^(s)_(t-)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A^(s)_(t-)群.为了深入研究有限非交换p-群,结合A^(s)_(t-)群的结构特点,本文描述具有A^(0)_(t-1)-子群的A_(t)-群的结构,并证明对于任意的A^(s)_(t-)群,若s>1,则s≡1(mod p),进一步地,完全分类所有的A1 t-群.
For a positive integer t,a finite p-group is called an A_(t)-group if all its subgroups of index p^(t)are abelian,but it has at least a non-abelian subgroup of index p^(t−1).An A_(t)-group is said to be an A^(s)_(t-)group if it has exactly s abelian subgroups of index p^(t−1),where s>0.For any finite non-abelian p-group,there must be suitable integers s and t such that it is an A^(s)_(t-)group.In order to study finite non-abelian p-groups deeply,combining with the structural characteristics of A^(s)_(t-)groups,we describe the structure of an A_(t)-group having an A^(0)_(t-1)-subgroup.We prove that for any A^(s)_(t-)group,if s>1,then s≡1(mod p),and completely classify the A 1 t-groups.
作者
白鹏飞
郭秀云
王俊新
Pengfei Bai;Xiuyun Guo;Junxin Wang
出处
《中国科学：数学》
CSCD
北大核心
2024年第2期139-160,共22页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:12171302和11801334)
山西省自然科学基金(批准号:202103021224287)
山西省高等学校科技创新项目(批准号:2021L278)资助项目。
