|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[group theory]], a '''locally cyclic group''' is a group (''G'', *) in which every [[generating set of a group|finitely generated subgroup]] is [[cyclic group|cyclic]].
| | Anybody who wrote the article is called Eusebio. His friends say it's negative for him but so what on earth he loves doing is going to be acting and he's have been doing it for quite a while. Filing has been his [http://www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=profession profession] although. Massachusetts has always been his everyday living place and his spouse and kids loves it. Go to his website to find out more: http://prometeu.net<br><br>Here is my web page ... hack clash of clans ([http://prometeu.net click through the up coming web page]) |
| | |
| ==Some facts==
| |
| *Every cyclic group is locally cyclic, and every locally cyclic group is [[abelian group|abelian]].
| |
| *Every finitely-generated locally cyclic group is cyclic.
| |
| *Every [[subgroup]] and [[quotient group]] of a locally cyclic group is locally cyclic.
| |
| *Every [[homomorphism|Homomorphic]] image of a locally cyclic group is locally cyclic.
| |
| *A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
| |
| *A group is locally cyclic if and only if its [[lattice of subgroups]] is [[distributive lattice|distributive]] {{harv|Ore|1938}}.
| |
| *The [[torsion-free rank]] of a locally cyclic group is 0 or 1.
| |
| | |
| ==Examples of locally cyclic groups that are not cyclic==
| |
| *The additive group of [[rational number]]s ('''Q''', +) is locally cyclic – any pair of rational numbers ''a''/''b'' and ''c''/''d'' is contained in the cyclic subgroup generated by 1/''bd''.
| |
| *The additive group of the [[dyadic rational number]]s, the rational numbers of the form ''a''/2<sup>''b''</sup>, is also locally cyclic – any pair of dyadic rational numbers ''a''/2<sup>''b''</sup> and ''c''/2<sup>''d''</sup> is contained in the cyclic subgroup generated by 1/2<sup>max(''b'',''d'')</sup>.
| |
| *Let ''p'' be any prime, and let μ<sub>''p''<sup>∞</sup></sub> denote the set of all ''p''th-power [[root of unity|roots of unity]] in '''C''', i.e.
| |
| | |
| :<math>\mu_{p^{\infty}} = \left\{ \exp\left(\frac{2\pi im}{p^{k}}\right) : m,k\in\mathbb{Z}\right\}</math> | |
| | |
| :Then μ<sub>''p''<sup>∞</sup></sub> is locally cyclic but not cyclic. This is the [[Prüfer group|Prüfer ''p''-group]]. The Prüfer 2-group is closely related to the dyadic rationals (it can be viewed as the dyadic rationals modulo 1).
| |
| | |
| ==Examples of abelian groups that are not locally cyclic==
| |
| *The additive group of [[real number]]s ('''R''', +) is not locally cyclic—the subgroup generated by 1 and π consists of all numbers of the form ''a'' + ''b''π. This group is [[group isomorphism|isomorphic]] to the [[direct sum of groups|direct sum]] '''Z''' + '''Z''', and this group is not cyclic.
| |
| | |
| ==References==
| |
| *{{citation
| |
| | last = Hall | first = Marshall, Jr. | author-link = Marshall Hall (mathematician)
| |
| | contribution = 19.2 Locally Cyclic Groups and Distributive Lattices
| |
| | isbn = 978-0-8218-1967-8
| |
| | pages = 340–341
| |
| | publisher = American Mathematical Society
| |
| | title = Theory of Groups
| |
| | year = 1999}}.
| |
| | |
| *{{citation
| |
| | last = Ore | first = Øystein | author-link = Øystein Ore
| |
| | doi = 10.1215/S0012-7094-38-00419-3
| |
| | mr = 1546048
| |
| | issue = 2
| |
| | journal = Duke Mathematical Journal
| |
| | pages = 247–269
| |
| | title = Structures and group theory. II
| |
| | volume = 4
| |
| | year = 1938}}.
| |
| | |
| [[Category:Abelian group theory]]
| |
Anybody who wrote the article is called Eusebio. His friends say it's negative for him but so what on earth he loves doing is going to be acting and he's have been doing it for quite a while. Filing has been his profession although. Massachusetts has always been his everyday living place and his spouse and kids loves it. Go to his website to find out more: http://prometeu.net
Here is my web page ... hack clash of clans (click through the up coming web page)