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Quasicyclic group

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 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}.