scholarly journals On a product of two formational tcc-subgroups

2020 ◽  
Vol 30 (2) ◽  
pp. 282-289
Author(s):  
A. Trofimuk ◽  
Keyword(s):  
Saturated Formation ◽  
Supersoluble Groups ◽  
Soluble Groups

A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G=AT and for any X≤A and Y≤T there exists an element u∈⟨X,Y⟩ such that XYu≤G. The notation H≤G means that H is a subgroup of a group G. In this paper we consider a group G=AB such that A and B are tcc-subgroups in G. We prove that G belongs to F, when A and B belong to F and F is a saturated formation of soluble groups such that U⊆F. Here U is the formation of all supersoluble groups.

scholarly journals On conditional permutability and saturated formations

2011 ◽  
Vol 54 (2) ◽  
pp. 309-319 ◽  
Author(s):  
M. Arroyo-Jordá ◽  
P. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Saturated Formation ◽  
Finite Product ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Permutable Subgroups ◽  
Saturated Formations

AbstractTwo subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

scholarly journals Languages associated with saturated formations of groups

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Adolfo Ballester-Bolinches ◽  
Jean-Éric Pin ◽  
Xaro Soler-Escrivà
Keyword(s):  
Group Theory ◽  
Nilpotent Groups ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Saturated Formations ◽  
Sylow Tower ◽  
General Method

AbstractIn a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.

On Finite Soluble Groups

1969 ◽  
Vol 9 (1-2) ◽  
pp. 250-251 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Normal Subgroup ◽  
Minimal Normal Subgroup ◽  
Saturated Formation ◽  
Fitting Subgroup ◽  
Minimal Order ◽  
Soluble Groups

Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g, F(G), the Fitting subgroup of G, is the unique minimal normal subgroup of G. It is to groups with this property that the following proposition is applicable.

Finite groups with given sets of 𝔉-subnormal subgroups

2019 ◽  
Vol 13 (04) ◽  
pp. 2050073 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Groups ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Hereditary Saturated Formation ◽  
Subnormal Subgroups

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

scholarly journals Fitting classes and lattice formations I

2004 ◽  
Vol 76 (1) ◽  
pp. 93-108 ◽  
Author(s):  
M. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Direct Product ◽  
Pairwise Disjoint ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Closure Properties ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
Disjoint Sets ◽  
Lattice Formation ◽  
Fitting Classes

AbstractA lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.

scholarly journals A Schunck class construction and a problem concerning primitive groups

1985 ◽  
Vol 38 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Peter Förster
Keyword(s):  
Finite Groups ◽  
Saturated Formation ◽  
Soluble Groups ◽  
Primitive Groups ◽  
Class Construction ◽  
Schunck Class ◽  
Usual Product

AbstractGaschütz has introduced the concept of a product of a Schunck class and a (saturated) formation (differing from the usual product of classes) and has shown that this product is a Schunck class provided that both of its factors consist of finite soluble groups. We investigate the same question in the context of arbitrary finite groups.

ON A PERMUTABILITY PROPERTY OF SUBGROUPS OF FINITE SOLUBLE GROUPS

2010 ◽  
Vol 12 (02) ◽  
pp. 207-221 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
JOHN COSSEY ◽  
X. SOLER-ESCRIVÀ
Keyword(s):  
Soluble Group ◽  
Subnormal Subgroup ◽  
Finite Soluble Group ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Subnormal Subgroups

The structure and embedding of subgroups permuting with the system normalizers of a finite soluble group are studied in the paper. It is also proved that the class of all finite soluble groups in which every subnormal subgroup permutes with the Sylow subgroups is properly contained in the class of all soluble groups whose subnormal subgroups permute with the system normalizers while this latter is properly contained in the class of all supersoluble groups.

An example in the theory of solubl groups

1970 ◽  
Vol 67 (1) ◽  
pp. 13-16 ◽  
Author(s):  
T. O. Hawkes
Keyword(s):  
Soluble Group ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Carter Subgroup ◽  
Soluble Groups ◽  
University Of Wisconsin ◽  
Unpublished Work ◽  
The University

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we provePROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

scholarly journals Finite soluble groups with certain splitting properties

1972 ◽  
Vol 14 (2) ◽  
pp. 237-246
Author(s):  
R. A. Bryce ◽  
John Cossey
Keyword(s):  
Normal Subgroup ◽  
Detailed Structure ◽  
Frattini Subgroup ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Structure Theorems

AbstractAbstract. In a recent paper, Bechtell obtained detailed structure theorems for finite supersoluble groups with the property that every minimal supplement for a non-Frattini normal subgroup is a complement. We consider finite soluble groups with this property. The situation is rather different to the supersoluble case, and the information we obtain is not as complete, though for such groups with non-trivial Frattini subgroup, some of the results are analogous.

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