fitting class
Recently Published Documents


TOTAL DOCUMENTS

42
(FIVE YEARS 0)

H-INDEX

6
(FIVE YEARS 0)

Author(s):  
Olesia V. Kamozina ◽  

All groups under consideration are assumed to be finite. For a nonempty subclass of Ω of the class of all simple groups I and the partition ζ = {ζi | i ∈ I}, where ζi is a nonempty subclass of the class I, I = ∪i∈I ζi and ζi ∩ ζj = ø for all i ≠ j, ΩζR-function f and ΩζFR-function φ are introduced. The domain of these functions is the set Ωζ ∪ {Ω′}, where Ωζ = { Ω ∩ ζi | Ω ∩ ζi ≠ ø }, Ω′ = I \ Ω. The scope of these function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions f and φ are used to determine the Ωζ-foliated Fitting class F = ΩζR(f, φ) = (G : OΩ(G) ∈ f(Ω′) and G'φ(Ω ∩ ζi) ∈ f(Ω ∩ ζi) for all Ω ∩ ζi ∈ Ωζ(G)) with Ωζ-satellite f and Ωζ-direction φ. The paper gives examples of Ωζ-foliated Fitting classes. Two types of Ωζ-foliated Fitting classes are defined: Ωζ-free and Ωζ-canonical Fitting classes. Their directions are indicated by φ0 and φ1 respectively. It is shown that each non-empty non-identity Fitting class is a Ωζ-free Fitting class for some non-empty class Ω ⊆ I and any partition ζ. A series of properties of Ωζ-foliated Fitting classes is obtained. In particular, the definition of internal Ωζ-satellite is given and it is shown that every Ωζ-foliated Fitting class has an internal Ωζ-satellite. For Ω = I, the concept of a ζ-foliated Fitting class is introduced. The connection conditions between Ωζ-foliated and Ωζ-foliated Fitting classes are shown.


Author(s):  
Martsinkevich Anna V.

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If  X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially soluble groups.


2018 ◽  
Vol 17 (12) ◽  
pp. 1850229 ◽  
Author(s):  
W. Guo ◽  
S. N. Vorob’ev

For a nonempty formation [Formula: see text], let [Formula: see text] be the least formation containing [Formula: see text] such that [Formula: see text] for all groups [Formula: see text] and [Formula: see text]. If [Formula: see text], then [Formula: see text] is said to be a DH-formation. A function [Formula: see text]: [Formula: see text] is called a formation function and [Formula: see text] is called the support of [Formula: see text]. If [Formula: see text] and [Formula: see text], where [Formula: see text], then [Formula: see text] is said to be a local formation defined by [Formula: see text]. If [Formula: see text] for all [Formula: see text], then [Formula: see text] is called a DH-function. In this paper, we prove that every local formation can be defined by a DH-function [Formula: see text] and each value of the DH-function [Formula: see text] is a DH-formation. Besides, we also give the necessary and sufficient condition under which a local Fitting class [Formula: see text] is a DH-formation.


2015 ◽  
Vol 22 (02) ◽  
pp. 211-214
Author(s):  
Yufeng Liu ◽  
Xiaolan Yi ◽  
N. T. Vorob'ev
Keyword(s):  

In this paper, we study the cover-avoid property of 𝔉-injectors on chief factors of a group G for a Fitting class 𝔉 in some universe 𝔘 with partial solubility.


2004 ◽  
Vol 76 (2) ◽  
pp. 175-188
Author(s):  
M. Arroyo-Jordá ◽  
M. D. Pérez-Ramos

AbstractGiven a lattice formation F of full characteristic, an F - Fitting class is a Fitting class with stronger closure properties involving F -subnormal subgroups. The main aim of this paper is to prove that the associated injectors possess a good behaviour with respect to F -subnormal subgroups.


2004 ◽  
Vol 76 (1) ◽  
pp. 23-38
Author(s):  
Stephanie Reifferscheid

AbstractIn the present paper we consider Fitting classes of finite soluble groups which locally satisfy additional conditions related to the behaviour of their injectors. More precisely, we study Fitting classes 1 ≠⊆such that an-injector of G is, respectively, a normal, (sub)modular, normally embedded, system permutable subgroup of G for all G ∈.Locally normal Fitting classes were studied before by various authors. Here we prove that some important results—already known for normality—are valid for all of the above mentioned embedding properties. For instance, all these embedding properties behave nicely with respect to the Lockett section. Further, for all of these properties the class of all finite soluble groups G such that an x-injector of G has the corresponding embedding property is not closed under forming normal products, and thus can fail to be a Fitting class.


2003 ◽  
Vol 68 (1) ◽  
pp. 107-112 ◽  
Author(s):  
J. C. Beidleman ◽  
H. Heineken

Dedicated to the memory of Bernhard H. NeumannNot often is a formation also a Fitting class. In this article groups are exhibited that are contained in a given saturated formation and that do not lead out of this formation by forming normal products with other groups of it. This generalises a result of Cossey on T-groups and supersolvable groups.


2002 ◽  
Vol 12 (5) ◽  
Author(s):  
O.V. Kamozina

AbstractWe describe the n-multiply Ω-bicanonical Fitting classes with Boolean lattice of Fitting subclasses. In particular, it is shown that in this case a Fitting class is directly decomposable with the use of the set of all atoms of its lattice. Here the notion of a direct decomposition plays the key role. Therefore we study direct decompositions separately and consider Ω-foliated Fitting classes with more general directions.


1996 ◽  
Vol 53 (2) ◽  
pp. 209-212
Author(s):  
Martin Menth

Metanilpotent Lockett classes are Hall closed. There is an example of a supersoluble, not Hall closed Fitting class.


Sign in / Sign up

Export Citation Format

Share Document