Quasicyclic group

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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 $(p n) t h$ 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.