Hamiltonian 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. 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.

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 2^{2a −6} quaternion groups as subgroups". In 2005 Horvat et al. used this structure to count the number of Hamiltonian groups of any order n = 2^eo 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), p. 190.

References

• Dedekind, Richard (1897), "Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind", Mathematische Annalen 48 (4): 548–561, doi:10.1007/BF01447922, ISSN 0025-5831, JFM 28.0129.03, MR1510943, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002256258 
• Hall, Marshall (1999), The theory of groups, AMS Bookstore, p. 190, ISBN 9780821819678 
• Horvat, Boris; Jaklič, Gašper; Pisanski, Tomaž (2005). "On the number of Hamiltonian groups". Mathematical Communications 10 (1): 89–94. 
• G. A. Miller (1898) "On the Hamilton groups", Bulletin of the American Mathematical Society 4(10):510–15.
• Olga Taussky-Todd (1970) "Sums of squares", American Mathematical Monthly: 77:805–30.