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.
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