I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in Syl(G)}\textbf{N}_G(S)$$
It is really easy if we use the theorem 3.v from the Baer's paper Group elements of prime power index. I wonder if there is a more direct way to prove this fact. I also tried to prove the next fact:
The hypercenter is the intersection of the maximal nilpotent subgroups.
I'm really stuck on these theorems provided that I can't figure out a satisfactory direct proof of them.