On Fisher Subgroups of Finite Groups

Let ℱ be a Fitting set of a group G, let π be a nonempty set of primes, and let L ≤ G. In this case, ℱ is called a Fischer π-set of G if the conditions that L ∈ ℱ, K ⊴ L, and H/K is a p-subgroup of L/K for a prime p ∈ π necessarily imply that H ∈ ℱ. It is said that a subgroup F of G is a Fischer ℱ-subgroup of G if the following assertions are true: (1) F ∈ ℱ; (2) if L is an ℱ-subgroup of G normalized by F, then L ≤ F. It is also said that a Fitting set ℱ of G is π-saturated if ℱ = {H ≤ G : H/H_ℱ ∈ 𝔈_π′}, where 𝔈_π′ is the class of all π′-groups. Under the condition that ℱ is a π-saturated Fischer π-set of a π-soluble group G, we prove that a subgroup V of G is an ℱ-injector of G if and only if V is a Fischer ℱ-subgroup of G containing a Hall π′-subgroup of G.