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388:. The Prüfer 2-group is closely related to the dyadic rationals (it can be viewed as the dyadic rationals modulo 1).
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362:{\displaystyle \mu _{p^{\infty }}=\left\{\exp \left({\frac {2\pi im}{p^{k}}}\right):m,k\in \mathbb {Z} \right\}}
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A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
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222:/2, is also locally cyclic – any pair of dyadic rational numbers
482:(1999), "19.2 Locally Cyclic Groups and Distributive Lattices",
230:/2 is contained in the cyclic subgroup generated by 1/2.
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Every finitely-generated locally cyclic group is cyclic.
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Examples of abelian groups that are not locally cyclic
191:, +) is locally cyclic – any pair of rational numbers
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Examples of locally cyclic groups that are not cyclic
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is contained in the cyclic subgroup generated by 1/(
486:, American Mathematical Society, pp. 340–341,
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136:image of a locally cyclic group is locally cyclic.
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43:but its sources remain unclear because it lacks
380:is locally cyclic but not cyclic. This is the
142:A group is locally cyclic if and only if its
129:of a locally cyclic group is locally cyclic.
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74:Learn how and when to remove this message
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407:, +); the subgroup generated by 1 and
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411:(comprising all numbers of the form
161:of a locally cyclic group is 0 or 1.
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508:"Structures and group theory. II"
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527:10.1215/S0012-7094-38-00419-3
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96:finitely generated subgroup
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214:The additive group of the
515:Duke Mathematical Journal
546:A Course on Group Theory
29:This article includes a
544:Rose, John S. (2012) .
216:dyadic rational numbers
58:more precise citations.
548:. Dover Publications.
437:, which is not cyclic.
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579:Properties of groups
574:Abelian group theory
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88:locally cyclic group
480:Hall, Marshall Jr.
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86:In mathematics, a
31:list of references
555:978-0-486-68194-8
493:978-0-8218-1967-8
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166:endomorphism ring
159:torsion-free rank
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467:Rose (2012)
455:Rose (2012)
170:commutative
134:homomorphic
56:introducing
568:Categories
442:References
428:direct sum
424:isomorphic
106:Some facts
347:∈
309:π
296:
278:∞
269:μ
250:th-power
64:June 2015
506:(1938),
152:Ore 1938
123:subgroup
535:1546048
426:to the
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258:, i.e.
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113:abelian
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100:cyclic
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