Guo, Wenbin; Vdovin, Evgeny P. Number of Sylow subgroups in finite groups. (English) Zbl 1404.20015 J. Group Theory 21, No. 4, 695-712 (2018). The authors denote by \(\nu_p(G)\) the number of Sylow \(p\)-subgroups of \(G\). It holds that \(\nu_p(H)\leq \nu_p(G)\) for \(h\leq G\), however \(\nu_p(H)\) does not divide \(\nu_p(G)\) in general. Let \(\mathrm{DivSyl}(p)\) for a finite group \(G\) stand for the property that \(\nu_p(H)\) divides \(\nu_p(G)\) for every subgroup \(H\) of \(G\). In this paper, the main result is the following: Theorem 1.1 (paraphrased). Suppose \(G\) does not satisfy \(\mathrm{DivSyl}(p)\). Then there exists a nonabelian composition factor of \(G\) not satisfying \(\mathrm{DivSyl}(p)\). Theorem 1.1 is an extension to a theorem of G. Navarro [Proc. Am. Math. Soc. 131, No. 10, 3019–3020 (2003; Zbl 1030.20010)], he showed Theorem 1.1 to be true for \(p\)-solvable groups \(G\). In order to prove Theorem 1.1, the authors formulate and prove several lemmas and theorems, most of them being rather detailed but useful in their own account.The paper is an enrichment to the vast already known properties of Sylow structures in a finite group. Reviewer: Robert W. van der Waall (Amsterdam) Cited in 1 Document MSC: 20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure 20D15 Finite nilpotent groups, \(p\)-groups 20D30 Series and lattices of subgroups 20D35 Subnormal subgroups of abstract finite groups 20D45 Automorphisms of abstract finite groups Keywords:finite groups; Sylow \(p\)-subgroups; \(p\)-solvable groups; composition series; chief series Citations:Zbl 1030.20010 PDFBibTeX XMLCite \textit{W. Guo} and \textit{E. P. Vdovin}, J. Group Theory 21, No. 4, 695--712 (2018; Zbl 1404.20015) Full Text: DOI arXiv References: [1] F. Gross, On the existence of Hall subgroups, J. Algebra 98 (1986), no. 1, 1-13. · Zbl 0588.20018 [2] M. Hall, Jr., On the number of Sylow subgroups in a finite group, J. Algebra 7 (1967), 363-371. · Zbl 0178.02102 [3] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. [4] I. M. Isaacs, Finite Group Theory, Grad. Stud. Math. 92, American Mathematical Society, Providence, 2008. · Zbl 1169.20001 [5] A. S. Kondrat’ev, Normalizers of Sylow 2-subgroups in finite simple groups, Math. Notes 78 (2005), no. 3-4, 338-346. · Zbl 1111.20017 [6] G. Navarro, Number of Sylow subgroups in p-solvable groups, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3019-3020. · Zbl 1030.20010 [7] A. Turull, The number of Hall π-subgroups of a π-separable group, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2563-2565. · Zbl 1045.20015 [8] E. P. Vdovin, Carter subgroups in finite almost simple groups, Algebra Logika 46 (2007), no. 2, 157-216. · Zbl 1155.20019 [9] E. P. Vdovin, Groups of induced automorphisms and their application for the study of the existence of Hall subgroups, Algebra Logika 53 (2014), no. 5, 643-648. [10] E. P. Vdovin and V. I. Zenkov, The intersection of soluble Hall subgroups in finite groups, Tr. Inst. Mat. Mech. UrO RAN 15 (2009), no. 2, 74-83. [11] V. A. Vedernikov, Finite groups with Hall π-subgroups, Mat. Sb. 203 (2012), no. 3, 23-48. · Zbl 1256.20014 [12] Z. Wu, W. Guo and E. P. Vdovin, On the number of Sylow subgroups in special linear groups of degree 2, Algebra Logic 56 (2017), no. 6, 498-501. · Zbl 1390.20020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.