On nontrivial factorisations of a one-generated local formation of finite groups

1992 ◽  
pp. 363-374 ◽  
Author(s):  
A. N. Skiba
Keyword(s):  
Finite Groups ◽  
Local Formation ◽  
Formation Of Finite Groups

scholarly journals Frattini classes of saturated formations of finite groups

1990 ◽  
Vol 42 (2) ◽  
pp. 267-286 ◽  
Author(s):  
Peter Förster
Keyword(s):  
Finite Groups ◽  
Local Formation ◽  
Formation Of Finite Groups ◽  
Saturated Formations ◽  
Definition Of

We study the following question: given any local formation of finite groups, do there exist maximal local subformations? An answer is given by providing a local definition of the intersection of all maximal local subformations.

scholarly journals Local definitions of local homomorphs and formations of finite groups

1985 ◽  
Vol 31 (1) ◽  
pp. 5-34 ◽  
Author(s):  
P. Förster ◽  
E. Salomon
Keyword(s):  
Finite Groups ◽  
Local Formation ◽  
Soluble Groups ◽  
Formation Of Finite Groups ◽  
Formation Function ◽  
Definition Of

It is well known that every local formation of finite soluble groups possesses three distinguished local definitions consisting of finite soluble groups: the minimal one, the full and integrated one, and the maximal one. As far as the first and the second of these are concerned, this statement remains true in the context of arbitrary finite groups. Doerk, Šemetkov, and Schmid have posed the problem of whether every local formation of finite groups has a distinguished (that is, unique) maximal local definition. In this paper a description of local formations with a unique maximal local definition is given, from which counter-examples emerge. Furthermore, a criterion for a formation function to be a local definition of a given local formation is obtained.

The lattice properties of 𝔛-local formations of finite groups

2020 ◽  
pp. 2150043 ◽  
Author(s):  
Aleksandr Tsarev
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Local Formation ◽  
Common Extension ◽  
Completeness Property

Let [Formula: see text] be a class of simple groups with a completeness property [Formula: see text]. Förster introduced the concept of [Formula: see text]-local formation in order to obtain a common extension of well-known theorems of Gaschütz–Lubeseder–Schmid and Baer [Publ. Mat. UAB 29(2–3) (1985) 39–76]. In this paper, it is proved that the lattice of all [Formula: see text]-local formations of finite groups is modular.

A generalized Frattini subgroup

2020 ◽  
pp. 2150026
Author(s):  
Ruslan V. Borodich
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
General Question ◽  
Frattini Subgroup ◽  
Local Formation ◽  
Glasgow Math ◽  
A Finite Group ◽  
Subnormal Subgroups

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

The formation of finite groups with a supersolvable π-hall subgroup

2010 ◽  
Vol 87 (1-2) ◽  
pp. 258-263 ◽  
Author(s):  
Xiaolan Yi ◽  
L. A. Shemetkov
Keyword(s):  
Finite Groups ◽  
Hall Subgroup ◽  
Formation Of Finite Groups

On $\mathfrak{F}_n$-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS

2008 ◽  
Vol 01 (04) ◽  
pp. 619-629 ◽  
Author(s):  
Nanying Yang ◽  
Wenbin Guo
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Formation Of Finite Groups ◽  
A Finite Group

Let G be a finite group and denote a formation of finite groups by [Formula: see text]. We call a subgroup H of G[Formula: see text]-supplemented in G if there exists a normal subgroup K of G such that G = HK and (H ∩ K)HG/HG is contained in the [Formula: see text]-hypercenter [Formula: see text] of G/HG. In this paper, we use [Formula: see text]-supplemented subgroups to study the structure of finite groups. Some known results in the literature are generalized.

scholarly journals SCHREIER CONDITIONS ON CHIEF FACTORS AND RESIDUALS OF SOLVABLE-LIKE GROUP FORMATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 97-106
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Solvable Groups ◽  
Fitting Subgroup ◽  
Group Extensions ◽  
Formation Of Finite Groups ◽  
Generalized Fitting Subgroup ◽  
Chief Factors ◽  
The Generalized Fitting Subgroup ◽  
Structural Characterizations

AbstractLet α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.

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