𝜔𝜎-fibered Fitting classes
The paper considers only finite groups. A class of groups F is called a Fitting class if it is closed under normal subgroups and products of normal F-subgroups; formation, if it is closed with respect to factor groups and subdirect products; Fitting formation if F is a formation and Fitting class at the same time. For a nonempty subset of the set of primes P and the partition = {𝜎𝑖 | ∈ 𝐼}, where P = ∪𝑖∈𝐼𝜎𝑖 and 𝜎𝑖∩𝜎𝑗 = ? for all 𝑖 ̸= 𝑗, we introduce the 𝜔𝜎𝑅-function and 𝜔𝜎𝐹𝑅-function 𝜙. The domain of these functions is the set 𝜔𝜎∪{𝜔′}, where = {𝜔∩𝜎𝑖 | 𝜔∩𝜎𝑖 ̸= ?}, 𝜔′ = P∖𝜔. The range of function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions and are used to determine the 𝜔𝜎-fibered Fitting class F = 𝜔𝜎𝑅(𝑓, 𝜙) = (𝐺 : 𝑂𝜔(𝐺) ∈ 𝑓(𝜔′) and 𝐺𝜙(𝜔∩𝜎𝑖) ∈ 𝑓(𝜔 ∩ 𝜎𝑖) for all ∩ ∈ 𝜔𝜎(𝐺)) with the 𝜔𝜎-satellite and the 𝜔𝜎-direction 𝜙. The paper gives examples of 𝜔𝜎-fibered Fitting classes. Two types of 𝜔𝜎-fibered Fitting classes are distinguished: 𝜔𝜎-complete and 𝜔𝜎-local Fitting classes. Their directions are indicated by 𝜙0 and 𝜙1, respectively. It is shown that each nonempty nonidentity Fitting class is an 𝜔𝜎-complete Fitting class for some nonempty set ⊆ P and any partition 𝜎. A number of properties of 𝜔𝜎-fibered Fitting classes are obtained. In particular, a definition of an internal 𝜔𝜎-satellite is given and it is shown that each 𝜔𝜎-fibered Fitting class always has an internal 𝜔𝜎-satellite. For = P, the concept of a 𝜎-fibered Fitting class is introduced. The connection between 𝜔𝜎-fibered and 𝜎-fibered Fitting classes is shown.