A note on compact elements of the lattice of solubly saturated formations

2015 ◽  
Vol 08 (04) ◽  
pp. 1550088
Author(s):  
Nikolay N. Vorob’ev
Keyword(s):  
Finite Groups ◽  
Saturated Formation ◽  
Compact Element ◽  
Saturated Formations ◽  
Solubly Saturated Formation

It is proved that every [Formula: see text]-closed solubly saturated formation contained in a compact element of the lattice of all [Formula: see text]-closed saturated formations of finite groups is also contained in some compact element of the lattice of all [Formula: see text]-closed solubly saturated formations 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.

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].

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

scholarly journals Schunck classes and projectors in a class of locally finite groups

1995 ◽  
Vol 38 (3) ◽  
pp. 511-522 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Finite Case ◽  
Saturated Formations ◽  
Definition Of ◽  
Schunck Class

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

ON SYLOW NORMALIZERS OF FINITE GROUPS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350116 ◽  
Author(s):  
L. S. KAZARIN ◽  
A. MARTÍNEZ-PASTOR ◽  
M. D. PÉREZ-RAMOS
Keyword(s):  
Finite Groups ◽  
Property A ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Hall Subgroups ◽  
Saturated Formations ◽  
The Universe ◽  
Do So

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

On Lattices of Formations of Finite Groups

2010 ◽  
Vol 17 (04) ◽  
pp. 557-564 ◽  
Author(s):  
L. A. Shemetkov ◽  
A. N. Skiba ◽  
N. N. Vorob'ev
Keyword(s):  
Finite Groups ◽  
Saturated Formations

Let ω be a set of primes with |ω| > 1, and m > n ≥ 0 be integers. It is proved that the lattice of all τ-closed m-multiply ω-saturated formations is not a sublattice of the lattice of all τ-closed n-multiply ω-saturated formations.

OnG-Covering Subgroup Systems for Some Saturated Formations of Finite Groups

2013 ◽  
Vol 41 (8) ◽  
pp. 2948-2956 ◽  
Author(s):  
Wenbin Guo ◽  
Jianhong Huang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Groups ◽  
Saturated Formations

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.

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