# Quasicyclic group

## 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.