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On Hua-Tuan’s conjecture

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Abstract

Let G be a finite group and |G| = p n, p be a prime. For 0 ⩽ m ⩽ n, s m (G) denotes the number of subgroups of of order p m of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then s m (G) ≡ 1, 1 + p, 1 + p + p 2 or 1 + p + 2p 2 (mod p 3). 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.

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Authors and Affiliations

  1. School of Mathematics and Computer Sciences, Shanxi Normal University, Linfen, 041004, China

    QinHai Zhang & HaiPeng Qu

Authors
  1. QinHai Zhang
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  2. HaiPeng Qu
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Correspondence to QinHai Zhang.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10671114), the Natural Science Foundation of Shanxi Province (Grant No. 2008012001) and the Returned Abroad-Student Fund of Shanxi Province (Grant No. [2007]13-56)

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Zhang, Q., Qu, H. On Hua-Tuan’s conjecture. Sci. China Ser. A-Math. 52, 389–393 (2009). https://doi.org/10.1007/s11425-009-0020-z

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  • DOI: https://doi.org/10.1007/s11425-009-0020-z

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