(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G be a finite group, P ∈ Sylp(G) and W ⊆ P. We say that W is weakly closed in P with respect to G if Wg ⊆ P. In this paper, we explore structural properties of finite groups using the transfer homomorphism and the notion of weak closure in Sylow subgroups. We establish that if a central subgroup H ≤ Z(G) has finite index [G : H] coprime to | H |, then G ∼=H × ker(v), where v : G → H is the transfer. Furthermore, we characterize weakly closed subgroups W ≤ P ∈ Sylp(G) as normal in both NG(P) and all Sylow p subgroups containing W. Several consequences concerning conjugacy and normality are discussed.