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...)
 
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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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The quasicyclic group is Abelian, locally finite, and locally cyclic, any two subgroups of it are comparable, and every subgroup is characteristic.

Revision as of 13:00, 7 October 2008

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.

The quasicyclic group is Abelian, locally finite, and locally cyclic, any two subgroups of it are comparable, and every subgroup is characteristic.