Ωζ-foliated Fitting Classes

Abstract

All groups under consideration are assumed to be finite. For a nonempty subclass of Ω of the class of all simple groups I and the partition ζ = {ζi | i ∈ I}, where ζi is a nonempty subclass of the class I, I = ∪i∈I ζi and ζi ∩ ζj = ø for all i ≠ j, ΩζR-function f and ΩζFR-function φ are introduced. The domain of these functions is the set Ωζ ∪ {Ω′}, where Ωζ = { Ω ∩ ζi | Ω ∩ ζi ≠ ø }, Ω′ = I \ Ω. The scope of these function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions f and φ are used to determine the Ωζ-foliated Fitting class F = ΩζR(f, φ) = (G : OΩ(G) ∈ f(Ω′) and G'φ(Ω ∩ ζi) ∈ f(Ω ∩ ζi) for all Ω ∩ ζi ∈ Ωζ(G)) with Ωζ-satellite f and Ωζ-direction φ. The paper gives examples of Ωζ-foliated Fitting classes. Two types of Ωζ-foliated Fitting classes are defined: Ωζ-free and Ωζ-canonical Fitting classes. Their directions are indicated by φ0 and φ1 respectively. It is shown that each non-empty non-identity Fitting class is a Ωζ-free Fitting class for some non-empty class Ω ⊆ I and any partition ζ. A series of properties of Ωζ-foliated Fitting classes is obtained. In particular, the definition of internal Ωζ-satellite is given and it is shown that every Ωζ-foliated Fitting class has an internal Ωζ-satellite. For Ω = I, the concept of a ζ-foliated Fitting class is introduced. The connection conditions between Ωζ-foliated and Ωζ-foliated Fitting classes are shown.

Keywords

• finite group
• fitting class
• ωζ-foliated
• ωζ-satellite
• ωζ-direction