On two questions from the Kourovka notebook
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.
scientific article; zbMATH DE number 6845895
scientific article