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.
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)
关键词
华罗庚
数学理论
数学家
推测方法
Hua-Tuan＇s conjecture, p-groups of maximal class, enumeration of subgroups, Magma package
