On $τ$-closed $n$-multiply $σ$-local formations of finite groups

Abstract

All groups under consideration are finite. Let σ = {σ i | i ∈ I} be some partition of the set of all primes P, G be a group, and F be a class of groups. Then σ(G)A function f of the form f : σ → {formations of groups} is called a formation σ-function. For any formation σ-function f the class LF σ (f ) is defined as follows:). If for some formation σ-function f we have F = LF σ (f ), then the class F is called σ-local and f is called a σ-local definition of F. Every formation is called 0-multiply σ-local. For n > 0, a formation F is called n-multiply σ-local provided either F = (1) is the class of all identity groups or F = LF σ (f ), where f (σ i ) is (n − 1)-multiply σ-local for all σ i ∈ σ(F). A formation is called totally σ-local if it is n-multiply σ-local for all nonnegative integer n.Let τ (G) be a set of subgroups of G such that G ∈ τ (G). Then τ is called a subgroup functor if for every epimorphism ϕ : A → B and any groups H ∈ τ (A) and T ∈ τ (B) we have H ϕ ∈ τ (B) andIn this paper, we describe some properties of τ -closed n-multiply σ-local formations of finite groups, as well as the main properties of the lattice of such formations. In particular, we prove that the set l τ σn of all τ -closed n-multiply σ-local formations forms a complete modular algebraic lattice of formations. In addition, we proof that the lattice l τ σn is σ-inductive and G-separable.If for some formation σ-function f the equality F = LF σ (f ), then the class F is called σ-local, and f called σ-local definition of F. We write 1) is the class of all identity groups or F = LF σ (f ), where f (σ i ) is (n − 1)-multiply σ-local for all σ i ∈ σ(F). A formation is called totally σ-local if it is n-multiply σ-local for all nonnegative integer n.