Editing Dedekind group

{{Short description|Group such that every its subgroup is normal.}}
In [[group theory]], a '''Dedekind group''' is a [[group (mathematics)|group]] ''G'' such that every [[subgroup]] of ''G'' is [[normal subgroup|normal]].
All [[abelian group]]s are Dedekind groups.
A non-abelian Dedekind group is called a '''Hamiltonian group'''.<ref>{{cite book|author=Hall |title=The theory of groups|year=1999|url={{Google books|plainurl=y|id=oyxnWF9ssI8C|page=190|text=Hamiltonian}}|page=190}}</ref>

The most familiar (and smallest) example of a Hamiltonian group is the [[quaternion group]] of order 8, denoted by Q<sub>8</sub>.
Dedekind and [[Reinhold Baer|Baer]] have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a [[direct product of groups|direct product]] of the form {{nowrap|1=''G'' = Q<sub>8</sub> × ''B'' × ''D''}}, where ''B'' is an [[elementary abelian 2-group]], and ''D'' is a [[torsion group|torsion]] 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 group]]s). He named the non-abelian ones after [[William Rowan Hamilton]], the discoverer of [[quaternion]]s.

In 1898 [[George Abram Miller|George Miller]] delineated the structure of a Hamiltonian group in terms of its [[order (group theory)|order]] and that of its subgroups. For instance, he shows "a Hamilton group of order 2<sup>''a''</sup> has {{nowrap|2<sup>2''a'' − 6</sup>}} quaternion groups as subgroups". In 2005 Horvat ''et al''<ref>{{cite arXiv|last1=Horvat|first1=Boris|last2=Jaklič|first2=Gašper|last3=Pisanski|first3=Tomaž|date=2005-03-09|title=On the Number of Hamiltonian Groups|eprint=math/0503183}}</ref> used this structure to count the number of Hamiltonian groups of any order {{nowrap|1=''n'' = 2<sup>''e''</sup>''o''}} 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''.

== Notes ==
{{Reflist}}

== References ==
* {{Citation | last1=Dedekind | first1=Richard | author1-link=Richard Dedekind | title=Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind | doi=10.1007/BF01447922 | mr=1510943 | jfm = 28.0129.03 | year=1897 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=48 | issue=4 | pages=548–561 | s2cid=119992274 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002256258}}.
* Baer, R. Situation der Untergruppen und Struktur der Gruppe, Sitz.-Ber. Heidelberg. Akad. Wiss.2, 12–17, 1933.
* {{Citation |title=The theory of groups |last=Hall |first=Marshall |author-link=Marshall Hall (mathematician) |year=1999 |publisher=AMS Bookstore |isbn=978-0-8218-1967-8 |page=190 }}.
* {{citation | last1 = Horvat | first1 = Boris | last2 = Jaklič | first2 = Gašper | last3 = Pisanski | first3 = Tomaž |author3-link=Tomaž Pisanski|year = 2005 | title = On the number of Hamiltonian groups | journal = Mathematical Communications | volume = 10 | issue = 1| pages = 89–94 | bibcode = 2005math......3183H | arxiv = math/0503183 }}.
*{{citation|first=G. A.|last=Miller|year=1898|title=On the Hamilton groups|journal= [[Bulletin of the American Mathematical Society]] |volume=4|issue=10|pages=510–515|doi=10.1090/s0002-9904-1898-00532-3|doi-access=free}}.
*{{citation|first=Olga|last=Taussky|author-link=Olga Taussky-Todd|year=1970|title=Sums of squares|journal=[[American Mathematical Monthly]]|volume= 77|issue=8|pages=805–830|mr=0268121|doi=10.2307/2317016|jstor=2317016|hdl=10338.dmlcz/120593|hdl-access=free}}.

[[Category:Group theory]]
[[Category:Properties of groups]]