Hamiltonian group
A Hamiltonian group is a non-abelian group in which every subgroup is normal (a nonabelian Dedekind group).
- Abelian groups also have every subgroup normal, but are not called Hamiltonian — the term requires nonabelian.
Dedekind's classification
G is Hamiltonian ⟺ G ≅ Q₈ × B × D, where
- Q₈ = quaternion group of order 8,
- B = elementary abelian 2-group (possibly trivial),
- D = abelian group in which every element has odd order.
Consequences
- The smallest Hamiltonian group is
Q₈, of order 8. - Every Hamiltonian group contains a copy of Q₈.
- Every Hamiltonian group is nilpotent of class 2.
Examples
Non-example: D₄, the dihedral group of order 8, is nonabelian but its order-2 reflection subgroups are not normal — so D₄ is not Hamiltonian.