On some properties of the lattice of totally σ-local formations of finite groups
Throughout this paper, all groups are finite. Let $σ=\{σ_i{}|i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, $G$ is a group, and $\mathfrak{F}$ is a class of groups, then $σ(n)=\{σ_i{}|σ_i{}\cap \pi(n)\ne \emptyset\}$, $σ(G)=σ(|G|)$, and $σ(\mathfrak{F})=\cup _G...
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| Format: | Article |
| Language: | Belarusian |
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Belarusian State University
2020-12-01
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| Series: | Журнал Белорусского государственного университета: Математика, информатика |
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| Online Access: | https://journals.bsu.by/index.php/mathematics/article/view/3444 |
| _version_ | 1818875865779404800 |
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| author | Vasilly G. Safonov Inna Nikolaevna Safonova |
| author_facet | Vasilly G. Safonov Inna Nikolaevna Safonova |
| author_sort | Vasilly G. Safonov |
| collection | DOAJ |
| description | Throughout this paper, all groups are finite. Let $σ=\{σ_i{}|i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, $G$ is a group, and $\mathfrak{F}$ is a class of groups, then $σ(n)=\{σ_i{}|σ_i{}\cap \pi(n)\ne \emptyset\}$, $σ(G)=σ(|G|)$, and $σ(\mathfrak{F})=\cup _G{}_\in{}_\mathfrak{F}σ(G)$. A function $f$ of the form $f\colon σ\to$ {formations of groups} is called a formation σ-function. For any formation $σ$-function $f$ the class $LF_σ(f)$ is defined as follows:
$LF_{\sigma}(f)=(G$ is a group $|G=1$ или $G\ne1$ and $G/O_σ{}'_i{}_,{}_σ{}_i{}(G)\in f(σ_i{})$ for all $σ_i{}\in σ(G))$.
If for some formation $σ$-function $f$ we have $\mathfrak{F}=LF_{\sigma}(f)$, then the class $\mathfrak{F}$ is called $σ$-local definition of $\mathfrak{F}$. Every formation is called 0-multiply $σ$-local. For $n$ > 0, a formation $\mathfrak{F}$ is called $n$-multiply $σ$-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f)$, where $f(σ_i{})$ is $(n – 1)$-multiply $σ$-local for all $σ_i{}\in σ(\mathfrak{F})$. A formation is called totally $σ$-local if it is $n$-multiply $σ$-local for all non-negative integer $n$. The aim of this paper is to study properties of the lattice of totally $σ$-local formations. In particular, we prove that the lattice of all totally $σ$-local formations is algebraic and distributive. |
| first_indexed | 2024-12-19T13:33:17Z |
| format | Article |
| id | doaj.art-329f32e599234e3d9344d7f1cd16e8ed |
| institution | Directory Open Access Journal |
| issn | 2520-6508 2617-3956 |
| language | Belarusian |
| last_indexed | 2024-12-19T13:33:17Z |
| publishDate | 2020-12-01 |
| publisher | Belarusian State University |
| record_format | Article |
| series | Журнал Белорусского государственного университета: Математика, информатика |
| spelling | doaj.art-329f32e599234e3d9344d7f1cd16e8ed2022-12-21T20:19:17ZbelBelarusian State UniversityЖурнал Белорусского государственного университета: Математика, информатика2520-65082617-39562020-12-01361610.33581/2520-6508-2020-3-6-163444On some properties of the lattice of totally σ-local formations of finite groupsVasilly G. Safonov0https://orcid.org/0000-0003-0682-3107Inna Nikolaevna Safonova1https://orcid.org/0000-0001-6896-7208Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, BelarusBelarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, BelarusThroughout this paper, all groups are finite. Let $σ=\{σ_i{}|i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, $G$ is a group, and $\mathfrak{F}$ is a class of groups, then $σ(n)=\{σ_i{}|σ_i{}\cap \pi(n)\ne \emptyset\}$, $σ(G)=σ(|G|)$, and $σ(\mathfrak{F})=\cup _G{}_\in{}_\mathfrak{F}σ(G)$. A function $f$ of the form $f\colon σ\to$ {formations of groups} is called a formation σ-function. For any formation $σ$-function $f$ the class $LF_σ(f)$ is defined as follows: $LF_{\sigma}(f)=(G$ is a group $|G=1$ или $G\ne1$ and $G/O_σ{}'_i{}_,{}_σ{}_i{}(G)\in f(σ_i{})$ for all $σ_i{}\in σ(G))$. If for some formation $σ$-function $f$ we have $\mathfrak{F}=LF_{\sigma}(f)$, then the class $\mathfrak{F}$ is called $σ$-local definition of $\mathfrak{F}$. Every formation is called 0-multiply $σ$-local. For $n$ > 0, a formation $\mathfrak{F}$ is called $n$-multiply $σ$-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f)$, where $f(σ_i{})$ is $(n – 1)$-multiply $σ$-local for all $σ_i{}\in σ(\mathfrak{F})$. A formation is called totally $σ$-local if it is $n$-multiply $σ$-local for all non-negative integer $n$. The aim of this paper is to study properties of the lattice of totally $σ$-local formations. In particular, we prove that the lattice of all totally $σ$-local formations is algebraic and distributive.https://journals.bsu.by/index.php/mathematics/article/view/3444finite groupformation σ-functionformation of finite groupstotally σ-local formationlattice of formations |
| spellingShingle | Vasilly G. Safonov Inna Nikolaevna Safonova On some properties of the lattice of totally σ-local formations of finite groups Журнал Белорусского государственного университета: Математика, информатика finite group formation σ-function formation of finite groups totally σ-local formation lattice of formations |
| title | On some properties of the lattice of totally σ-local formations of finite groups |
| title_full | On some properties of the lattice of totally σ-local formations of finite groups |
| title_fullStr | On some properties of the lattice of totally σ-local formations of finite groups |
| title_full_unstemmed | On some properties of the lattice of totally σ-local formations of finite groups |
| title_short | On some properties of the lattice of totally σ-local formations of finite groups |
| title_sort | on some properties of the lattice of totally σ local formations of finite groups |
| topic | finite group formation σ-function formation of finite groups totally σ-local formation lattice of formations |
| url | https://journals.bsu.by/index.php/mathematics/article/view/3444 |
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