Remarks on formations

Applying the normalizer theory of finite groups developed in 1989, we undertake some questions concerning the theory of formations of finite groups.     

A note on saturated formations

This work is part of the Proyecto PS 87-0055-C02-02 of CAICYT (Ministerio de Educación y Ciencia, Spain) and has been done during a visit of the author to the Fachbereich Mathematik of the Johannes Gutenberg-Universität Mainz (West Germany). The author wants to thank this Institution. Special thanks are due to Prof. Dr. K. Doerk for the discussions which led to this paper.     

Radical formations

A formation is called radical (weakly n-radical) if it contains every group G which can be represented in the formG=M ₁M₂ ...M _n,M _iG, where the subgroups M_i belong to (belong to and have pairwise prime indices). It is proved that a local formation is radical (weakly n-radical,n 2) if and only if its complete inner local screen f has the following property: f(p) is a radical (a weakly n-radical,n 2) formation for every prime number p.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 861–864, June, 1977.     

p-Saturated formations

We consider here a nonempty formation F locally defined by a system {F(t)} where for some fixed prime numberp, F(p) = F. We call such formationsp-locally defined. It is shown that everyp-locally defined formation F has the property thatG/Φ _p (G ∈ F) implies thatG ∈ F where Φ _p (G) is thep-Frattini subgroup ofG. If F is a formation of solvable groups, then this property is equivalent to F having ap-local definition. For solvable groups equivalent conditions are stated in terms of the F-projectors, the F-normalizers, and the F-hypercenter.     

SolvableF-radical formations-radical formations

In the paper we list all solvable normally hereditary local formationsF such that in any finite groupG the product of every commutingF-subnormal subgroups ofG is aF-subnormal subgroup of the groupG.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 261–266, February, 1996.     

Factorizations of composition formations

It is proved that, if is a singly generated composition formation, where , then is a composition formation. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 389–395, March, 1999.     

A question on saturated formations of finite groups

This research has been supported by Proyecto PB 90-0414-C03-03 of DGICYT, Ministerio de Educación y Ciencia of Spain. The second author is also supported by Proyecto Acciones de grupos finitos of Gobierno de Navarra, Spain.     

Soluble totally local formations

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 4, pp. 862–872, July–August, 1995.     

On formations like ℸℏ

Gomel. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 5, pp.181–187, September–October, 1993.     

On hereditary superradical formations

A formation F is superradical provided that: (1) F is a normally hereditary formation; (2) each group G = AB, where A and B are F-subnormal F-subgroups in G, belongs to F. We give an example of a hereditary superradical formation that is not soluble saturated. This gives a negative answer to Problem 14.99(b) in The Kourovka Notebook.