On Hua-Tuan's conjecture

作     者：ZHANG QinHai & QU HaiPeng School of Mathematics and Computer Sciences, Shanxi Normal University, Linfen 041004, China 

作者机构：1. School of Mathematics and Computer Sciences Shanxi Normal University Linfen 041004 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2009年第52卷第2期

页      面：389-393页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant No. 10671114) the Natural Science Foundation of Shanxi Province (Grant No. 2008012001) the Returned Abroad-Student Fund of Shanxi Province (Grant No. 13-56) 

主　　题：Hua-Tuan’s conjecture metacyclic p-groups abelian p-groups inner abelian p- groups superspecial p-groups 

摘      要：Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have 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). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.