On σ-local Fitting classes

Abstract

Let σ be a partition of the set of all primes P. If G is a finite group and F is a Fitting class of finite groups, the symbol σ(G) denotes the set {σi|σi∩π(|G|)≠∅} and σ(F)=∪σ∈Fσ(G). We call any function f of the form f:σ⟶{Fitting classes} a Hartley σ-function (or simply Hσ-function), and we put LRσ(f)=(G|G=1orG≠1andGG∈f(σi)for allσi∈σ(G)). If there is an Hσ-function f such that F=LRσ(f), then we say that F is σ-local and f is a σ-local definition of F. In this paper, we describe some properties of σ-local Fitting classes and prove that: 1) every σ-local Fitting class can be defined by a unique Hσ-function F such that F(σi)=F(σi)Gσ⊆F and F(σi) is a Lockett class for all σi∈σ(F); 2) the product of two σ-local Fitting classes is also a σ-local Fitting class. Moreover, we also discuss the n-multiply σ-local Fitting classes.