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.