×

On the \(\Pi\)-property of subgroups of finite groups. (English) Zbl 1335.20022

Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is said to satisfy the \(\Pi\)-property in \(G\) if every prime dividing \(|G:N_G(HK\cap L)|\) also divides the order of \((HK\cap L)/K\), for every chief factor \(L/K\) of \(G\).
In this note the authors prove that the finite group \(G\) is soluble if and only if all maximal subgroups of \(G\) satisfy the \(\Pi\)-property in \(G\). This answers affirmatively the Question 5.2 posed by B. Li [in J. Algebra 334, No. 1, 321-337 (2011; Zbl 1248.20020)].

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Citations:

Zbl 1248.20020
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, volume 584 of Mathematics and its Applications, Springer, New York, 2006. · Zbl 1102.20016
[2] J. H. Conway et al. Atlas of Finite Groups, Oxford Univ. Press, London, 1985. · Zbl 1134.20021
[3] Jiménez-Seral P.: Coefficients of the probabilistic function of a monolithic group. Glasg. Math. J., 50, 75-81 (2008) · Zbl 1134.20021 · doi:10.1017/S0017089507004053
[4] Li B.: On \[{{\Pi}}\] Π-property and \[{{\Pi}}\] Π-normality of subgroups of finite groups. J. Algebra, 334, 321-337 (2011) · Zbl 1248.20020 · doi:10.1016/j.jalgebra.2010.12.018
[5] Li C., Li X.: On permutation groups of degree a product of two prime-powers. Comm. Algebra, 42, 4722-4743 (2014) · Zbl 1304.20002 · doi:10.1080/00927872.2013.823500
[6] M. W. Liebeck, C. E. Praeger, and J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra, 111(1987), 365-383. · Zbl 0632.20011
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.