This is probably standard for group-theorists:
Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center?
If so, where do I find a proof? What about the same question for infinite groups?


The original question can be reformulated as follows: Let $G$ be a finite group and fix $g\in G$. Assume that for every $x\in G$ there exists a natural number $k(x)$ such that
$$
gxg^{-1}=x^{k(x)},
$$
does it follow that $k(x)=1$ for all $x$? One gets the reformulation by applying the original statement to cyclic groups. It suffices to consider cyclic groups, as an element that normalizes all cyclic groups, normalizes every subgroup.



Intersection of all normalizers
No. There are non-abelian groups $G$ for which all subgroups are normal, such as the quaternion group of order 8. So the intersection of all normalizers is just $G$.


