Difference between revisions of "Quasicyclic group"
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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]]. | | [[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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| + | ==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]]. | ||
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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
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 
.
