On n-multiply σ-local formations of finite groups

Abstract

Throughout this paper, all groups are finite. Let be some partition of the set of all primes . If n is an integer, the symbol denotes the set ; and . We call any function f of the forma formation σ-function, and we putIf for some formation σ-function f we have , then we say that the class is σ-local and f is a σ-local definition of . We suppose that every formation is 0-multiply σ-local; for n > 0, we say that the formation is n-multiply σ-local provided either is the class of all identity groups or where is multiply σ-local for all . In this paper, we describe some properties and examples of n-multiply σ-local formations. In particular, we prove that the Gaschütz product of any two n-multiply σ-local formations is also n-multiply σ-local. We also consider one application of such formations in the theory of finite factorizable groups.