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Finite p-Groups Whose Length of Chain of Nonnormal Subgroups is At Most 2

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Abstract

Assume that G is a finite non-Dedekind p-group, where p is an odd prime. Passman introduced the following concept: we say that \(H_{1}<H_{2}<\cdots <H_{k}\) is a chain of nonnormal subgroups of G if each \(H_{i}\ntrianglelefteq G\) and if \(|H_i:H_{i-1}|=p\) for \(i=2,3,\dots , k\). k is called the length of the chain. \(\mathrm{chn}(G)\) denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite p-groups G with \(\mathrm{chn}(G)\le 2\) are completely classified up to isomorphism.

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Acknowledgements

The author cordially thanks the referee for detailed reading and valuable comments. The author thanks Professor Qinhai Zhang for his helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Shanxi Normal University, Linfen, Shanxi, 041004, People’s Republic of China

    Qiangwei Song

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  1. Qiangwei Song
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Correspondence to Qiangwei Song.

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Communicated by Hamid Mousavi.

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This work was supported by NSFC (Nos. 11901367 and 11771258).

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Song, Q. Finite p-Groups Whose Length of Chain of Nonnormal Subgroups is At Most 2. Bull. Iran. Math. Soc. 46, 737–754 (2020). https://doi.org/10.1007/s41980-019-00288-2

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