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On finite groups with non-nilpotent subgroups

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Abstract

For a finite group \(G\), let \(l(G)\) denote the number of conjugacy classes of non-normal non-nilpotent subgroups of \(G\). In this paper, we show that every finite group \(G\) satisfying \(l(G)< |\pi (G)|\) is solvable, and for a finite non-solvable group \(G\), \(l(G)=|\pi (G)|\) if and only if \(G\cong A_5\) or \(SL(2,5)\).

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Acknowledgments

The authors are grateful to the referee who gives valuable comments and suggestions. J. Lu is supported by the National Natural Science Foundation of China (11461007, 11261007) and by the Guangxi Natural Science Foundation Program (2013GXNSFBA019003, 2014GXNSFAA118009). W. Meng is partially supported by the National Natural Science Foundation of China (11361075).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, Guangxi, People’s Republic of China

    Jiakuan Lu

  2. School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, 650031, Yunnan, People’s Republic of China

    Wei Meng

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  1. Jiakuan Lu
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  2. Wei Meng
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Correspondence to Jiakuan Lu.

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Communicated by J. S. Wilson.

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Lu, J., Meng, W. On finite groups with non-nilpotent subgroups. Monatsh Math 179, 99–103 (2016). https://doi.org/10.1007/s00605-014-0712-5

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  • DOI: https://doi.org/10.1007/s00605-014-0712-5

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