A criterion for σ-locality of a non-empty formation. Issue 6 (3rd June 2022)

• Record Type: Journal Article
• Authors: Safonova, Inna N.
• Is Part Of: Communications in algebra. Volume 50:Issue 6(2022)
• Page Start: 2366
• Page End: 2376
• Publication Date: 2022-06-03
• Subjects: Class of finite groups -- formation σ-function -- generated formation -- n-multiply σ-local formation -- σ-local formation; 20F17 -- 20D10; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.2006210
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: 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 ) = { 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 | G = 1 or G ≠ 1 and G / O σ i ′, σ i ( G ) ∈ f ( σ i ) for all σ i ∈ σ ( G ) ) . If 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 ) . We describe new properties of n -multiply σ -local formations. In particular, we give conditions under which a non-empty formation is n -multiply σ -local.