On Weakly Conjugate ζ-Permutable Subgroups of Finite Groups

Let C ≠ϕ be subset of G and ζ represents complete set of Sylow subgroups of G. Let Ԋ ≤ G is C- ζ permutable (conjugate ζ permutable) subgroup within G if for ԊxGq= GqԊx ∃ x∈C, ∀ Gq ∈ ζ be a finite group. A subgroup Ԋ≤G is known as weakly Conjugate ζ-permutable within G when ∃, ????≤ G s.t G= Ԋ???? and Ԋ∩????≤Ԋcζ where <Ԋcζ>⊆Ԋ are conjugate ζ permutable within G. Our main goal: G is supersolvable when maximal subgroups of Gq∩Ƒ∗(G) are weakly C-ζ-permutable within G, for each Gq ∈ ζ, where Ƒ∗(G) denote generalized Fitting subgroup of G. Moreover, we show when, G∈ Ƒ, Ƒ is saturated formation consist of every supersolvable groups if and only if Ԋ⊴G s.t G/Ԋ∈ Ƒ and maximal subgroups of Gq∩Ƒ∗(Ԋ) becomes weakly C- ζ-permutable Within G, for each Gq∈ ζ.