Hamiltonian group

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In group theory, a Dedekind group is a group G such that every subgroup of G is normal. All abelian groups are Dedekind groups. A non-abelian Dedekind group is called a Hamiltonian group.

The most familiar (and smallest) example of a Hamiltonian group is the quaternion group of order 8, denoted by Q₈. It can be shown that every Hamiltonian group is a direct product 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.

Dedekind groups are named after Richard Dedekind, who investigated them in (Dedekind 1897), proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after William Rowan Hamilton, the discoverer of quaternions.

[edit] References

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• Dedekind, Richard (1897), "Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind", Mathematische Annalen 48 (4): 548–561, doi:10.1007/BF01447922, MR1510943 JFM 28.0129.03 &format=complete 28.0129.03 , ISSN 0025-5831