Local Fitting sets and the injectors of a finite group
The product F ◊X of the Fitting set F of a group G and the Fitting class X is called the set of subgroups {H ≤ G: H/HF ∈ X}. Let P be the set of all primes, ∅ ≠ π ⊆ P, π′ = Pπ and Eπ′ denote the class of all π′-groups. Let S and Sπ to denote the class of all soluble groups and the class of all π -soluble groups, respectively. In the paper, it is proved that F-injector of a group G either covers or avoids every chief factor of G if G is a partially soluble group. Chief factors of a group covered by F-injectors are described in the following cases: 1) G ∈ F ◊ S and F is the Hartley set of G; 2) G ∈ Sπ and F = F ◊ Eπ′ for the integrated H-function f.