Editing Locally cyclic group

{{no footnotes|date=June 2015}}
In mathematics, a '''locally cyclic group''' is a [[group (mathematics)|group]] (''G'', *) in which every [[finitely generated subgroup]] is [[cyclic group|cyclic]].

==Some facts==
* Every cyclic group is locally cyclic, and every locally cyclic group is [[abelian group|abelian]].{{sfnp|Rose|2012|p=54}}
* Every finitely-generated locally cyclic group is cyclic.
* Every [[subgroup]] and [[quotient group]] of a locally cyclic group is locally cyclic.
* Every [[group 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.
* The [[endomorphism ring]] of a locally cyclic group is [[commutative ring|commutative]].{{citation needed|date=June 2015}}

==Examples of locally cyclic groups that are not cyclic==
{{unordered list
| 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'').{{sfnp|Rose|2012|p=52}}
| 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 ''&mu;''<sub>''p''<sup>&infin;</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''', +); the subgroup generated by 1 and {{pi}} (comprising all numbers of the form ''a''&nbsp;+&nbsp;''b''{{pi}}) is [[group isomorphism|isomorphic]] to the [[direct sum of groups|direct sum]] '''Z''' + '''Z''', which is not cyclic.

== See also ==
* [[Bézout domain]]

==References==
{{Reflist}}

{{Refbegin}}
*{{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| url = http://dml.cz/bitstream/handle/10338.dmlcz/100155/CzechMathJ_05-1955-3_8.pdf
 }}.

*{{cite book|title=A Course on Group Theory|last=Rose|first=John S.|date=2012|publisher=Dover Publications|isbn=978-0-486-68194-8|orig-date=unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978}}
{{Refend}}

[[Category:Abelian group theory]]
[[Category:Properties of groups]]