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Harmonic mean Sylow numbers of nonsolvable groups

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Abstract

Let G be a finite group. We write hsn(G) for the harmonic mean Sylow number of G. The Fitting subgroup of G is denoted by F(G). It is known that if \(hsn(G)<\frac{45}{7 }\), then G is solvable. In this paper, we extend the study to nonsolvable groups. We prove that if G is a finite nonsolvable group, then the following holds:

  1. (a)

    if \(hsn(G)<\frac{360}{47}\) and \(hsn(G)\ne \frac{480}{71}\), then \( G/F(G)\cong A_{5}\);

  2. (b)

    if \(hsn(G)=\frac{480}{71}\), then \(G/N\cong A_{5}\), where N is the largest normal solvable subgroup of G.

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References

  1. C. S. Anabanti, A. Moretó and M. Zarrin, Influence of the number of Sylow subgroups on solvability of finite groups, Comptes Rendus Mathématique 358(11–12) (2020), 1227–1230.

    MathSciNet  Google Scholar 

  2. J. H. Conway, R. T. Curtis, S. P. Norton and R. A. Wilson, Atlas of finite groups, Clarendon, Oxford, 1985.

    Google Scholar 

  3. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.12.2; 2022, https://www.gap-system.org.

  4. Z. Habibi and M. Hezarjaribi, Some criteria for solvability and supersolvability, Journal of Mahani Mathematical Research; https://doi.org/10.22103/JMMR.2023.20047.1315

  5. M. Hall, The Theory of Groups, Macmillan, New York, 1959.

    Google Scholar 

  6. M. Hall, On the number of Sylow subgroups in a finite group, J. Algebra 7 (1967), 363–371.

    Article  MathSciNet  Google Scholar 

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Correspondence to C. S. Anabanti.

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Communicated by B. Sury.

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Anabanti, C.S., Asboei, A.K. Harmonic mean Sylow numbers of nonsolvable groups. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00617-0

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  • DOI: https://doi.org/10.1007/s13226-024-00617-0

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