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 { σ i | σ i ∩ π ( | G | ) ≠ ∅ } and σ ( F ) = ∪ σ ∈ F σ ( G ) . We call any function f of the form f : σ ⟶ { Fitting classes } a Hartley σ-function (or simply H σ -function), and we put L R σ ( f ) = ( G | G = 1 or G ≠ 1 and G G σ i G σ i ′ ∈ f ( σ i ) for all σ i ∈ σ ( G ) ) . If there is an H σ -function f such that F = L R σ ( 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 H σ -function F such that F ( σ i ) = F ( σ i ) G σ i ⊆ F and F ( σ i ) is a Lockett class for all σ i ∈ 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.