Are all nilpotent groups hamiltonian? That is, is every subgroup of a nilpotent group normal?
I don't think so. Every Sylow subgroup of nilpotent group is normal and every nilpotent group is a direct product of its Sylow subgroups. But, are they hamiltonian?
And, would the converse be true? That is, is every hamiltonian group nilpotent? Any small counterexamples? Thanks beforehand.