Hamilton group

Hamiltonian group

A non-Abelian group all subgroups of which are normal (cf. Normal subgroup). Such groups were studied by R. Dedekind and were named "Hamiltonian groups" by him after the creator of the algebra of quaternions, W.R. Hamilton. A non-Abelian group is Hamiltonian if and only if it is a direct product of the quaternion group of order 8, an Abelian group each element of which is of finite odd order, and an Abelian group of exponent 1 or 2. In particular, any Hamiltonian group is periodic (cf. Periodic group).

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