On <i>n</i>-multiply <i>σ</i>-local formations of finite groups

Abstract

Throughout this paper, all groups are finite. Let σ={σi|i∈I} be some partition of the set of all primes P. If n is an integer, the symbol σ(n) denotes the set {σi|σi∩π(n)≠∅}; σ(G)=σ(|G|) and σ(F)=∪G∈Fσ(G). We call any function f of the form f:σ→{formations of groups} a formation σ-function, and we put LFσ(f)=(G is a group|G=1 or G≠1 and G/Oσi′,σi(G)∈f(σi) for all σi∈σ(G)). If for some formation σ-function f we have F=LFσ(f), then we say that the class F is σ-local and f is a σ-local definition of F. We suppose that every formation is 0-multiply σ-local; for n > 0, we say that the formation F is n-multiply σ-local provided either F=(1) is the class of all identity groups or F=LFσ(f), where f(σi) is (n−1)-multiply σ-local for all σi∈σ(F). 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.