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.

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.

p-adics: inverse limit instead of direct limit
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 $$\mathbb{Z}/p^{n-1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$$.