Fitting Classes with Given Properties of Hall Subgroups

In the theory of formations of finite solvable groups, there is a well-known result due to Blessenohl claiming that, for any local formation , the class of groups for which every Hall π-subgroup belongs to also is a local formation. In the present paper, we obtain a result exactly dual to that indicated in the theory of Fitting classes. We prove that if a Fitting class is local, then the class of all groups all of whose Hall π-subgroups belong to is also local.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 234–240.Original Russian Text Copyright © 2005 by V. N. Zagurskii, N. T. Vorob’ev.     

On Modularity of the Lattice of Totally Saturated Formations of Finite Groups

It is proved that the lattice of totally saturated formations of finite groups is modular.     

Complete Distributivity of Lattice Ordered Groups and of Vector Lattices

In this paper we investigate the possibility of a regular embedding of a lattice ordered group into a completely distributive vector lattice.     

格序群的C-根类和K-根类

本文研究了Ｃ-根类（即凸ｌ－子群格可分辨根类）和Ｋ－根类．证明了：投射ｌ－群、强投射ｌ－群、两两分裂ｌ－群、Ｆ－群等许多ｌ－群类都是ｌ－子群格可分辨的,但Ｈａｍｉｌｔｏｎｌ－群不是;Ｃ－根类格是根类格的一个有伪补的闭子格,而且是根类半群的一个子半群．每个凸Ｏ一子群都是闭的,而且给出了由一簇Ｋ一根类及一个根类所生成的Ｋ－根类的形式．     

两个Fitting类的例子

给出了两个特殊的群类,即R={H|C_H(F(H))≤F(H)}和S={G|1≠K⊿G~((∞)),G~((∞))=KC_(G(∞))(K)},证明了他们都是Fitting类,这两个Fitting类都和单群理论有密切联系.各自都包含了一些很值得注意的群,对每个Fitting类给出了群G属于该Fitting类的一个充要条件.     

Products of Fitting Classes of Finite Groups

Fitting classes are considered. In particular, it is shown that a product of n-multiply ω-local Fitting classes is an n-multiply ω-local Fitting class.     

On Structural and Asymptotic Properties of Some Classes of Distributions

We consider the two-parametric classes of distributions which comprise exponential dispersion models possessing a simple structure. Some of our models are related to Galton-Watson processes and branching diffusions. We study asymptotic properties of related classes of distributions emphasizing their convergence to Poisson-exponential law. We establish/refute infinite divisibility for certain classes of distributions. The critical value for the deflation of zeros from a geometric law which preserves infinite divisibility is determined. A class of distributions related to zero-modified geometric and logarithmic laws is considered.     

关于格的卡氏积的特征多项式和几何性

利用格L_i(i=1,2)的性质研究了它们的卡氏积L=L_1×L_2的性质,得到了L的秩函数、Mbius函数和特征多项式,并且由L_i的几何性证明了L的几何性.     

Classes of Discrete Convexity Properties

In the general first-level classification of the convexity properties for sets, discrete convexities appear in more classes. A second-level classification identifies more subclasses containing discrete convexity properties, which appear as approximations either of classical convexity or of fuzzy convexity. First, we prove that all these convexity concepts are defined by segmental methods. The type of segmental method involved in the construction of discrete convexity determines the subclass to which it belongs. The subclasses containing the convexity properties that have discrete particular cases are also presented.     

$$\omega $$ -Fibered Formations and Fitting Classes of Finite Groups

In this paper, we suggest a new functional approach to the study of group classes which enables us to describe all formations and Fitting classes of finite groups in the language of functions. The notions of -fibered formation and of -fibered Fitting class with direction are introduced. A direction is defined as a mapping of the set of all primes into the set of all nonempty Fitting formations. The existence of infinitely many mappings of this kind makes it possible to construct new forms of formations and Fitting classes for a given nonempty set . In particular, an -local formation is an -fibered formation with a direction such that for any prime p. In the paper we study some basic properties of -fibered formations and of -fibered Fitting classes with direction and obtain the structure of their minimal satellites for a given .     

Distributivity conditions and the order-skeleton of a lattice

We introduce “π-versions” of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D _0π if aù(búc)  \leqslant  (aùb)úc{{{a}\wedge({b}\vee{c})\;\leqslant\;({a}\wedge{b})\vee{c}}} for all 3-element antichains {a, b, c}. We consider a congruence relation ~ whose blocks are the maximal autonomous chains and define the order-skeleton of a lattice L to be [(L)\tilde] : = L/ ~ {{\tilde{L} : = L/{\sim}}}. We prove that the following are equivalent for a lattice L: (i) L satisfies D _0π , (ii) [(L)\tilde]{{\tilde{L}}} satisfies any of the five π-versions of distributivity, (iii) the order-skeleton [(L)\tilde]{{\tilde{L}}} is distributive.     

Martin's Axiom and the Dual Distributivity Number

We show that it is consistent that Martin's axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ₁. This answers a question of Halbeisen.     

Approximation Properties and Some Classes of Operators

The following questions and close problems are studied.(i) Is it true that T is p-nuclear provided that T ^** is p-nuclear? (ii) Is it true that Tis dually p-nuclear provided that T ^* is p-nuclear? (iii) Is it true that if T ^*is compactly factorable in the space l _p, then T is (strictly) factorable in the space l _p'? Here, T ^* is the adjoint operator of a bounded operator T:X Yin Banach spaces X and Y. Bibliography: 30 titles.     

P-集合的格论性质

P-集合是把动态特性引入到有限普通集合中,改进普通集合得到的,P-集合是由内P-集合(internal packet sets)与外P-集合(outer packet sets)构成的集合对.利用P-集合理论,给出只集合的依赖关系和度量,证明P-集合在依赖关系下构成格,进而给出基于格论的P-.集合的相关性质.     

Lattice Properties of Torsion Theories

The main purpose of this brief note is to show that the sets of all hereditary torsion theories which are noetherian, strongly noetherian, or of finite type, respectively, form a sublattice of the lattice of all hereditary torsion theories for the category R-mod.     

On the Isomorphism Classes of Transversals

In this article we prove that there does not exist a subgroup H of a finite group G such that the number of isomorphism classes of right transversals of H in G is two.