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.
References
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This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (July 2009) |
- Dedekind, Richard (1897), "Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind", Mathematische Annalen 48 (4): 548–561, doi:, MR1510943 JFM 28.0129.03, ISSN 0025-5831
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