Hamiltonian group

In group theory, a non-abelian group G is called Hamiltonian if every subgroup H of G is also a normal subgroup of G.

Clearly, every abelian group has this property, because all subgroups of an abelian group are normal subgroups, but there are non-abelian examples as well. The most familiar is the quaternion group of order 8.

It can be shown that every Hamiltonian group is a direct sum of the form G = Q₈ + B + D, where Q₈ is the quaternion group of order 8, B is the direct sum of some number of copies of the cyclic group C₂, and D is a periodic abelian group with all elements of odd order.