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 *Q*_{8}. It can be shown that every Hamiltonian group is a direct product of the form *G* = *Q*_{8} × *B* × *D*, where *B* is the direct sum of some number of copies of the cyclic group *C*_{2}, 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.

In 1898 George Miller delineated the structure of a Hamiltonian group in terms of its order and that of its subgroups. For instance, he shows "a Hamilton group of order 2*a* has 22*a* −6 quaternion groups as subgroups". In 2005 Horvat *et al.* used this structure to count the number of Hamiltonian groups of any order *n* = 2e*o* where *o* is an odd integer. When *e* ≤ 3 then there are no Hamiltonian groups of order *n*, otherwise there are the same number as there are Abelian groups of order *o*.

### Famous quotes containing the word group:

“Once it was a boat, quite wooden

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—Anne Sexton (1928–1974)