On two questions from the Kourovka notebook
type: Item
instance of: scholarly article
title: On two questions from the Kourovka notebook
zbMATH Open document ID: 1428.20020
author: Adolfo Ballester-Bolinches
author: John Cossey
author: S. F. Kamornikov
author: Hangyang Meng
published in: Journal of Algebra
publication date: 2018-03-01T00:00:00Z
full work available at URL: http://hdl.handle.net/1885/139099
review text: The paper discussess the following two open questions in [\textit{V. D. Mazurov} (ed.) and \textit{E. I. Khukhro} (ed.), The Kourovka notebook. Unsolved problems in group theory. 17th ed. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2010; Zbl 1211.20001)]: Question 1 [loc. cit., Problem 17.55]: Does there exist an absolute constant $k$ such that the Frattini subgroup $\Phi(G)$ of a soluble group $G$ is the intersection of $k$ $G$-conjugates of any prefrattini subgroup $H$ of $G$? Question 2 [loc. cit., Problem 17.39]: Is there a positive integer $k$ such that the hypercentre of any finite soluble group coincides with the intersection of $k$ system normalisers of that group? What is the least number with this property? The authors give a complete answer to a general version of Question 1. It provides a beautiful description of the intersection of four (in some cases three) $\mathcal{F}$-prefrattini subgroups of a soluble group introduced by \textit{T. Hawkes} [in: Proceedings of the international conference on the theory of groups held at the Australian National University Canberra, 10--20 August, 1965. New York-London-Paris: Gordon and Breach Science Publishers. 145--150 (1967; Zbl 0183.03002)], where $\mathcal{F}$ is a saturated formation. The authors also obtain a result which implies the answer of Question 2 for groups with trivial Frattini subgroup.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 6845895
zbMATH Keywords: finite groups
zbMATH Keywords: soluble groups
zbMATH Keywords: formations
zbMATH Keywords: Fitting classes
zbMATH Keywords: prefrattini subgroups
zbMATH Keywords: normalisers
zbMATH Keywords: injectors
reviewed by: Yangming Li
MaRDI profile type: Publication
OpenAlex ID: W2780628229
cites work: Classes of Finite Groups
cites work: Finite soluble groups
cites work: Large orbits in coprime actions of solvable groups
cites work: Über die \(\Phi\)-Untergruppe endlicher Gruppen
cites work: Pre-Frattini groups.
cites work: Q5574015
cites work: Intersections of prefrattini subgroups in finite soluble groups
cites work: Addendum to "The Intersection of Sylow Subgroups"
cites work: Q3084921
cites work: On large orbits of subgroups of linear groups
cites work: Groups with Normal, Solvable Hall p � -Subgroups
cites work: Q4396576
DOI: 10.1016/J.JALGEBRA.2017.12.014
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description: scientific article; zbMATH DE number 6845895