On some properties of the lattice of totally σ-local formations of finite groups

Abstract

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.