具有几乎链条件的有限p-群  

Finite p-groups with an almost chain condition

作　　者：白鹏飞 郭秀云[2] 王俊新[1] Pengfei Bai;Xiuyun Guo;Junxin Wang

机构地区：[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

出　　处：《中国科学：数学》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。

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.

关 键 词：有限P-群 A_(t)-群 A^(s)_(t)- 群 链条件 

分 类 号：O152.1[理学—数学]