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Encyclopedia > Hamiltonian group

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

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 (and smallest) is the quaternion group of order 8, denoted by Q8.

It can be shown that every Hamiltonian group is a direct sum of the form G = Q8 + B + D, where 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.

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