On the Lattice of F-Dnormal Subgroups in Finite Soluble Groups

Carter and Gaschütz theories beyond soluble groups

“…and N is non-abelian, N is a π-group and, consequently, M π , H π ∈ Hall π (G). Hence, there exists x ∈ G such that M x π = H π and M and H are conjugate by Lemma 4.4 (2).…”

“…[8,17], [9, Chapter V], [5,Chapters 4,6]). In [2] a definition of F-normality in soluble groups consistent with the lattice properties of F-subnormal subgroups is achieved. This concept is applied in this paper to extend Carter subgroups taking heed of its very definition, as nilpotent self-normalizing subgroups, and enables us to address the lack of a Carter and Gaschütz counterpart in the extension of Hall theory from soluble groups to π-separable groups.…”

Injectors in $$\pi $$-Separable Groups

“…We observe that N π is a saturated formation and appeal to the concept of N π -Dnormal subgroup. We refer to [1,2] for the concept of G-Dnormal subgroup for general saturated formations G, and specialize this definition to our particular saturated formation N π next. For notation, if ρ is a set of primes and G is a group, Hall ρ (G) denotes the set of all Hall ρ-subgroups of G. If p is a prime, then Syl p (G) stands for the set of all Sylow p-subgroups of…”

Extension of Carter subgroups in $π$-separable groups

“…[2, Definition 3.1] Let G be a non-empty saturated formation and let G be a group. A subgroup H of G is said to be G-Dnormal in G if π(|G : H|) ⊆ Char(G), and for every p ∈ Char(G) it holds that[H p G , H g(p) ] ≤ H,where g denotes the smallest local definition of G as local formation, and…”