Research ArticleNo Access

On the number of Sylow subgroups in finite simple groups

School of Science, Jiangnan University, Wuxi 214122, P. R. China

https://doi.org/10.1142/S0219498821501152Cited by:0 (Source: Crossref)

Abstract

Denote by $ν_{p} (G)$ $ν_{p} (G)$ the number of Sylow $p$ $p$ -subgroups of $G$ $G$ . For every subgroup $H$ $H$ of $G$ $G$ , it is easy to see that $ν_{p} (H) \leq ν_{p} (G)$ $ν_{p} (H) \leq ν_{p} (G)$ , but $ν_{p} (H)$ $ν_{p} (H)$ does not divide $ν_{p} (G)$ $ν_{p} (G)$ in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group $G$ satisfies DivSyl(p) if $ν_{p} (H)$ divides $ν_{p} (G)$ for every subgroup $H$ of $G$ . In this paper, we show that “almost for every” finite simple group $S$ , there exists a prime $p$ such that $S$ does not satisfy DivSyl(p).

Communicated by E. Vdovin

Keywords:

AMSC: 20D10, 20D15, 20D20

References

1. W. Guo and E. P. Vdovin , Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712. Crossref, Web of Science, Google Scholar
2. P. B. Kleidman and M. Liebeck , The Subgroup Structure of the Finite Classical Groups (Cambridge Univerity Press, Cambridge, 1990). Crossref, Google Scholar
3. J. Nagura , On the interval containing at least one prime number, Proc. Japan Acad. 28(4) (1952) 177–181. Crossref, Google Scholar
4. G. Navarro , Number of Sylow subgroups in $p$ -solvable groups, Proc. Amer. Math. Soc. 131(10) (2003) 3019–3020. Crossref, Web of Science, Google Scholar
5. L. Weisner , On the Sylow subgroups of the symmetric and alternating groups, Amer. J. Math. 47 (1925) 121–124. Crossref, Google Scholar
6. Zh. Wu, W. Guo and E. P. Vdovin , On the number of Sylow subgroups in special linear groups of degree 2, Algebra Logic 56(6) (2017) 498–501. Crossref, Web of Science, Google Scholar
7. K. Zsigmondy , Zur theorie der potenzreste, Monatsh. Math. 3(1) (1892) 265–284. Crossref, Google Scholar