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A Lower Bound on the Number of Maximal Subgroups in a Finite Group

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Abstract

For a finite group G, let m(G) denote the set of maximal subgroups of G and \(\pi (G)\) denote the set of primes which divide |G|. In this paper, we prove a lower bound on |m(G)| when G is not nilpotent, that is, \(|m(G)| \ge | \pi (G)| + p\), where \(p \in \pi (G)\) is the smallest prime that divides |G| such that the Sylow p- subgroup of G is not normal in G.

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References

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Acknowledgements

J. Lu is supported by the National Natural Science Foundation of China (11861015), the Innovation Project of Guangxi Graduate Education (JGY2019003), the Promotion Project of Basic Ability for Young and Middle-aged Teachers in Universities of Guangxi Province (2018KY0087), and the Training Program for 1,000 Young and Middle-aged Cadre Teachers in Universities of Guangxi Province. S. Wang is supported by the Innovation Project of Guangxi Graduate Education(XYCSZ2019086). W. Meng is supported by the National Natural Science Foundation of China (11761079).

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Correspondence to Jiakuan Lu.

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Communicated by Hamid Mousavi.

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Lu, J., Wang, S. & Meng, W. A Lower Bound on the Number of Maximal Subgroups in a Finite Group. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-019-00345-w

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Keywords

  • Finite group
  • Maximal group

Mathematics Subject Classification

  • 20D10
  • 20D20