Difference between revisions of "Quasicyclic group"

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(New page: ==Definition== Let <math>p</math> be a prime number. The <math>p</math>-quasicyclic group is defined in the following equivalent ways: * It is the group, under multiplication, of all...)
 
(p-adics: inverse limit instead of direct limit)
 
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{{prime-parametrized particular group}}
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==Definition==
 
==Definition==
  
Let <math>p</math> be a [[prime number]]. The <math>p</math>-quasicyclic group is defined in the following equivalent ways:
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Let <math>p</math> be a [[prime number]]. The '''<math>p</math>-quasicyclic group''' is defined in the following equivalent ways:
  
 
* It is the group, under multiplication, of all complex <math>(p^n)^{th}</math> roots of unity for all <math>n</math>.
 
* It is the group, under multiplication, of all complex <math>(p^n)^{th}</math> roots of unity for all <math>n</math>.
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where the maps are multiplication by <math>p</math> maps.
 
where the maps are multiplication by <math>p</math> maps.
  
The quasicyclic group is Abelian, locally finite, and locally cyclic, any two subgroups of it are comparable, and any two proper nontrivial subgroups are related by an automorphism.
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==Particular cases==
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{| class="sortable" border="1"
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! Prime number <math>p</math> !! <math>p</math>-quasicyclic group
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|-
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| 2 || [[2-quasicyclic group]]
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|-
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| 3 || [[3-quasicyclic group]]
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|}
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==Group properties==
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{| class="sortable" border="1"
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! Property !! Satisfied? !! Explanation !! Corollary properties satisfied
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|-
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| [[satisfies property::abelian group]] || Yes ||  || Hence, it is also a [[nilpotent group]] and a [[solvable group]].
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|-
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| [[satisfies property::locally cyclic group]] || Yes || ||
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|-
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| [[satisfies property::locally finite group]] || Yes || ||
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|-
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| [[satisfies property::p-group]] || Yes || || Hence, it is an [[satisfies property::abelian p-group]], so also a [[satisfies property::nilpotent p-group]].
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|}
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==Related notions==
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===Combining quasicyclic groups for all primes===
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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]].
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===p-adics: inverse limit instead of direct limit===
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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>.

Latest revision as of 02:11, 11 August 2012

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups

Definition

Let p be a prime number. The p-quasicyclic group is defined in the following equivalent ways:

  • It is the group, under multiplication, of all complex (p^n)^{th} roots of unity for all n.
  • It is the quotient L/\mathbb{Z} where L is the group of all rational numbers that can be expressed with denominator a power of p.
  • It is the direct limit of the chain of groups:

\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2\mathbb{Z} \to \dots \to \mathbb{Z}/p^n\mathbb{Z} \to .

where the maps are multiplication by p maps.

Particular cases

Prime number p p-quasicyclic group
2 2-quasicyclic group
3 3-quasicyclic group

Group properties

Property Satisfied? Explanation Corollary properties satisfied
abelian group Yes Hence, it is also a nilpotent group and a solvable group.
locally cyclic group Yes
locally finite group Yes
p-group Yes Hence, it is an abelian p-group, so also a nilpotent p-group.

Related notions

Combining quasicyclic groups for all primes

The restricted external direct product of the p-quasicyclic groups for all prime numbers p is isomorphic to \mathbb{Q}/\mathbb{Z}, 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 p-adics are constructed as an inverse limit for surjective maps \mathbb{Z}/p^n\mathbb{Z} \to \mathbb{Z}/p^{n-1}\mathbb{Z}, the quasicyclic group is constructed as a direct limit for injective maps \mathbb{Z}/p^{n-1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}.