Ωζ-foliated Fitting Classes
Ωζ-расслоенные классы Фиттинга

Olesia V. Kamozina, Bryansk State Technological University of Engineering
2020 Izvestiya of Saratov University New Series Series Mathematics Mechanics Informatics  
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
more » ... 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.
doi:10.18500/1816-9791-2020-20-4-424-433 fatcat:5w5iuhy7pjfnnmkgjacmrnzy5y