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 torsion 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
edit- ^ Hall (1999). The theory of groups. p. 190.
- ^ Horvat, Boris; Jaklič, Gašper; Pisanski, Tomaž (2005-03-09). "On the Number of Hamiltonian Groups". arXiv:math/0503183.
References
edit- 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, MR 1510943, S2CID 119992274.
- Baer, R. Situation der Untergruppen und Struktur der Gruppe, Sitz.-Ber. Heidelberg. Akad. Wiss.2, 12–17, 1933.
- Hall, Marshall (1999), The theory of groups, AMS Bookstore, p. 190, ISBN 978-0-8218-1967-8.
- Horvat, Boris; Jaklič, Gašper; Pisanski, Tomaž (2005), "On the number of Hamiltonian groups", Mathematical Communications, 10 (1): 89–94, arXiv:math/0503183, Bibcode:2005math......3183H.
- Miller, G. A. (1898), "On the Hamilton groups", Bulletin of the American Mathematical Society, 4 (10): 510–515, doi:10.1090/s0002-9904-1898-00532-3.
- Taussky, Olga (1970), "Sums of squares", American Mathematical Monthly, 77 (8): 805–830, doi:10.2307/2317016, hdl:10338.dmlcz/120593, JSTOR 2317016, MR 0268121.