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Quasicyclic group

824 bytes added, 02:11, 11 August 2012
p-adics: inverse limit instead of direct limit
| [[satisfies property::p-group]] || Yes || || Hence, it is an [[satisfies property::abelian p-group]], so also a [[satisfies property::nilpotent p-group]].
==Related notions==
===Combining quasicyclic groups for all primes===
The [[restricted external direct product]] of the <math>p</math>-quasicyclic groups for all prime numbers <math>p</math> is isomorphic to <math>\mathbb{Q}/\mathbb{Z}</math>, the [[group of rational numbers modulo integers]].
===p-adics: inverse limit instead of direct limit===
The [[additive group of p-adic integers]] can, in a vague sense, be considered to be constructed using a method dual to the method used to the [[quasicyclic group]]. While the <math>p</math>-adics are constructed as an inverse limit for surjective maps <math>\mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}/p^{n-1}\mathbb{Z}</math>, the quasicyclic group is constructed as a direct limit for injective maps <math>\mathbb{Z}/p^{n-1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}</math>.
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