# Difference between revisions of "Quasicyclic 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.