关于Sylow子群皆交换的p²q³阶群的构造

关于Sylow子群皆交换的p 2 q 3 阶群的构造 On the Structures of Groups of Order p 2 q 3 with Abelian Sylow Subgroups 陈松良 CHEN Songliang 1 first-author 贵州师范学院数学与计算机科学学院 贵州师范学院数学与计算机科学学院 School of Mathematics and Computer Science,Guizhou Normal College School of Mathematics and Computer Science,Guizhou Normal College 设p,q为奇素数,且p>q.本文对Sylow子群皆交换的p2q 3阶群进行了完全分类并获得了其全部构造:1)当q（p2-1）且p（q2+q+1）时,G恰有6个不同构的类型;2)当q（p-1）但p|（q2+q+1）时,G恰有8个不同构的类型;3)当q|（p-1）但q2（p-1）且p（q2+q+1）时,G恰有q2+19个不同构的类型;4)当q|（p-1）且p|（q2+q+1）但q2（p-1）时,G恰有q2+21个不同构的类型;5)当q2|（p-1）但q3（p-1）时,G恰有2q2+q+24个不同构的类型;6)当q3|（p-1）时,G恰有（q3+5q2+2q+52）/2个不同构的类型;7)当q|（p+1）但q2（p+1）时,G恰有10个不同构的类型;8)当q2|（p+1）但q3（p+1）时,G恰有12个不同构的类型;9)当q3|（p+1）时,G恰有13个不同构的类型. Let p,q be odd primes such that p>q,and G be groups of order p2q3 with Abelian Sylow subgroups.In this paper,it is discussed that the isomorphic classification of G,and their structures are completely determined.We have showed that: 1) If q doesn’t divide（p2-1） and p doesn’t divide（q2+q+1）,G has 6 nonisomorphic structures;2) If q doesn’t divide（p-1） and p divides（q2+q+1）,G has 8 nonisomorphic structures;3) If q divides（p-1） and q2 doesn’t divide（p-1） and p doesn’t divide（q2+q+1）,G has（q2+19） nonisomorphic structures;4) If q divides（p-1） and q2 doesn’t divide（p-1） and p divides（q2+q+1）,G has（q2+21） nonisomorphic structures;5) If q2 divides（p-1） and q3 doesn’t divide（p-1）,G has 2q2+q+24 nonisomorphic structures;6) If q3 divides（p-1）,G has（q3+5q2+2q+52）/2 nonisomorphic structures;7) If q divides（p+1） and q2 doesn’t divide（p+1）,G has 10 nonisomorphic structures;8) If q2 divides（p+1） and q3 doesn’t divide（p+1）,G has 12 nonisomorphic structures;9) If q3 divides（p+1）,G has 13 nonisomorphic structures. 有限群 同构分类 群的构造 finite group isomorphic classification structure of group 10.14188/j.1671-8836.2013.03.014 O152.1 贵州省自然科学基金资助项目（2010GZ77391） 2013-03-01 2021-04-01 3 59