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.
Best Answer
In addition to what has been answered: a subgroup $H$ of $G$ is called subnormal if there exists a series of subgroups $H=H_0 \lhd H_1 \lhd \cdots \lhd H_s=G$. A normal subgroup is obviously subnormal, but the converse is not true. Now, finite nilpotent groups are exaclty the finite groups in which every subgroup is subnormal. For a proof: see for example M.I. Isaacs, Finite Group Theory, Lemma 2.1.