Let be a prime number. The -quasicyclic group is defined in the following equivalent ways:
- It is the group, under multiplication, of all complex roots of unity for all .
- It is the quotient where is the group of all rational numbers that can be expressed with denominator a power of .
- It is the direct limit of the chain of groups:
where the maps are multiplication by maps.
The quasicyclic group is Abelian, locally finite, and locally cyclic, any two subgroups of it are comparable, and any two proper nontrivial subgroups are related by an automorphism.