# Dedekind group

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.[1]

The most familiar (and smallest) example of a Hamiltonian group is the quaternion group of order 8, denoted by Q8. Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form G = Q8 × B × D, where B is an elementary abelian 2-group, 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 2a has 22a − 6 quaternion groups as subgroups". In 2005 Horvat et al[2] used this structure to count the number of Hamiltonian groups of any order n = 2eo 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.

## Notes

1. ^ Hall (1999). The theory of groups. p. 190.
2. ^ Horvat, Boris; Jaklič, Gašper; Pisanski, Tomaž (2005-03-09). "On the Number of Hamiltonian Groups". arXiv:math/0503183.