Hartley Sets and Injectors of a Finite Group

Vorob`ev, N. T. Hartley sets and injectors of a finite group / N. T. Vorob`ev, T. B. Karaulova // Mathematical Notes. – 2019. – Vol. 105, № 1-2. – Р. 204–215.By a Fitting set of a group G one means a nonempty set of subgroups F of a finite group G which is closed under taking normal subgroups, their products, and conjugations of subgroups. In the present paper, the existence and conjugacy of F-injectors of a partially π-solvable group G is proved and the structure of F-injectors is described for the case in which F is a Hartley set of G

On sublattices of the lattice of all ω-composition formations of finite groups

It is proved that the lattice of all ω-local formations is a complete sublattice of the lattice of all ω-composition formations of finite groups

О модулярности решетки бэровских σ-локальных формаций

Throughout this paper, all groups are finite. A group class closed under taking homomorphic images and finite subdirect products is called a formation. The symbol σ denotes some partition of the set of all primes. V. G. Safonov, I. N. Safonova, A. N. Skiba (Commun. Algebra. 2020. Vol. 48, № 9. P. 4002–4012) defined a generalized formation σ-function. Any function f of the form f : σ È {Ø} → {formations of groups}, where f(Ø) ≠ ∅, is called a generalized formation σ-function. Generally local formations or so-called Baer-σ-local formations are defined by means of generalized formation σ-functions. The set of all such formations partially ordered by set inclusion is a lattice. In this paper it is proved that the lattice of all Baerσ-local formations is algebraic and modular.Все рассматриваемые группы конечны. Формацией называется класс групп, замкнутый относительно взятия гомоморфных образов и подпрямых произведений. Символом σ обозначают некоторое разбиение множества всех простых чисел. В работе В. Г. Сафонова, И. Н. Сафоновой, А. Н. Скибы (Commun. Algebra. 2020. Vol. 48, № 9. P. 4002–4012) определена обобщенная формационная σ-функция как отображение f : σ È {Ø} → {формации групп}, где f(Ø) ≠ ∅. При помощи обобщенной формационной σ-функции определены обобщенно локальные формации – так называемые бэровские σ-локальные формации. Множество всех таких формаций образует решетку по включению. В настоящей работе установлены свойства алгебраичности и модулярности этой решетки

The Bell Theorem as a Special Case of a Theorem of Bass

The theorem of Bell states that certain results of quantum mechanics violate inequalities that are valid for objective local random variables. We show that the inequalities of Bell are special cases of theorems found ten years earlier by Bass and stated in full generality by Vorob'ev. This fact implies precise necessary and sufficient mathematical conditions for the validity of the Bell inequalities. We show that these precise conditions differ significantly from the definition of objective local variable spaces and as an application that the Bell inequalities may be violated even for objective local random variables.Comment: 15 pages, 2 figure