# 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 $(p^n)^{th}$ 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.

## 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.

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