| 词条 | Dedekind group |
| 释义 |
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 {{nowrap|1=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 {{harv|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 {{nowrap|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 {{nowrap|1=n = 2eo}} where o is an odd integer. When {{nowrap|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. Notes1. ^{{cite book|author=Hall |title=The theory of groups|year=1999|url={{Google books|plainurl=y|id=oyxnWF9ssI8C|page=190|text=Hamiltonian}}|page=190}} 2. ^{{cite arxiv|last=Horvat|first=Boris|last2=Jaklič|first2=Gašper|last3=Pisanski|first3=Tomaž|date=2005-03-09|title=On the Number of Hamiltonian Groups|eprint=math/0503183}} References
2 : Group theory|Properties of groups |
| 随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。