On separability of the lattice of τ-closed n-multiply σ-local formations

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, F be a class of groups, σ ( G ) = { σ i | σ i ∩ π ( G ) ≠ ∅ } , and σ ( F ) = ∪ G ∈ F 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 L F σ ( f ) is defined as follows: L F σ ( f ) = ( G is a group |G = 1 or G ≠ 1 and G/O σ i ′ , σ i ( G ) ∈ f ( σ i ) for all σ i ∈ σ ( G ) ) . If for some formation σ-function f we have F = L F 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 = L F σ ( f ) , where f ( σ i ) is ( n − 1 ) -multiply σ-local for all σ i ∈ σ ( F ) . 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 ) and T φ − 1 ∈ τ ( A ) . A formation of groups F is called τ-closed if τ ( G ) ⊆ F for all G ∈ F . A complete lattice of formations θ is called separable, if for any term ν ( x 1 , … , x m ) signatures { ∩ , ∨ θ } , any θ-formations F 1 , … , F m and any group A ∈ ν ( F 1 , … , F m ) there are groups A 1 ∈ F 1 , … , A m ∈ F m such that A ∈ ν ( θ form A 1 , … , θ form A m ) . We prove that the lattice of all τ-closed n-multiply σ-local formations is a separable lattice of formations.