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| On Hua-Tuan’s conjecture | |
| 基金项目： | 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. [2007]13-56) |
| 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. |
| 关 键 词： | Hua-Tuan’s conjecture metacyclic p-groups abelian p-groups inner abelian p- groups superspecial p-groups |
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