Let $G$ be a finite group. TFAE:
(1) $G$ is nilpotent.
(2) $N_{G}(H) > H$ for all subgroups $H< G$.
(3) Every maximal subgroup of $G$ is a normal subgroup.
The proof of these equivalences seems to imply that $(1) \Rightarrow (2) \Rightarrow (3)$ holds for infinite groups as well. I was wondering if $(2)$ and/or $(3)$ are also equivalent with $(1)$ for infinite groups. I would have to work with nonabelian infinite nilpotent groups, but to be honest, I don't know any, which in addition would make thinking about normalizers and maximal subgroups harder.
Thanks.