# 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}} | ||

+ | |||

==Definition== | ==Definition== | ||

− | Let <math>p</math> be a [[prime number]]. The <math>p</math>-quasicyclic group is defined in the following equivalent ways: | + | 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. | ||

− | + | ==Particular cases== | |

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Prime number <math>p</math> !! <math>p</math>-quasicyclic group | ||

+ | |- | ||

+ | | 2 || [[2-quasicyclic group]] | ||

+ | |- | ||

+ | | 3 || [[3-quasicyclic group]] | ||

+ | |} | ||

+ | |||

+ | ==Group properties== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Property !! Satisfied? !! Explanation !! Corollary properties satisfied | ||

+ | |- | ||

+ | | [[satisfies property::abelian group]] || Yes || || Hence, it is also a [[nilpotent group]] and a [[solvable group]]. | ||

+ | |- | ||

+ | | [[satisfies property::locally cyclic group]] || Yes || || | ||

+ | |- | ||

+ | | [[satisfies property::locally finite group]] || Yes || || | ||

+ | |- | ||

+ | | [[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>. |

## 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

## Contents

## Definition

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

- It is the group, under multiplication, of all complex roots of unity for all .
- It is the quotient where is the group of all rational numbers that can be expressed with denominator a power of .
- It is the direct limit of the chain of groups:

.

where the maps are multiplication by maps.

## Particular cases

Prime number | -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 -quasicyclic groups for all prime numbers is isomorphic to , 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 -adics are constructed as an inverse limit for surjective maps , the quasicyclic group is constructed as a direct limit for injective maps .