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

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?	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 $\mathbb {Z} /p^{n-1}\mathbb {Z} \to \mathbb {Z} /p^{n}\mathbb {Z}$ .