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 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). The theory of groups. 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, MR 1510943.
• 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.
• 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: 805–830, doi:10.2307/2317016, MR 0268121.