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?	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.