On Hua-Tuan's conjecture Ⅱ  

作　　者：ZHANG QinHai QU HaiPeng 

机构地区：[1]Department of Mathematics, Shanxi Normal University, Linfen 041004, China

出　　处：《Science China Mathematics》2011年第1期65-74,共10页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.11071150);Natural Science Foundation of Shanxi Province (Grant No. 2008012001);The Returned Abroad-student Foundation of Shanxi Province (Grant No. [2007]13-56)

摘　　要：Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1+p, 1+p+p2 or 1+p+2p2(mod p3). The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.Let G be a group of order pn, p a prime. For 0 m n, sm（G） denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured： for an arbitrary finite p-group G, if p 〉 2, then sm（G） ≡ 1, 1＋p, 1＋p＋p2 or 1＋p＋2p2（mod p3）. The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.

关 键 词：Hua-Tuan＇s conjecture p-groups of maximal class enumeration of subgroups Magma package 

分 类 号：K826.11[历史地理—历史学]