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 every subgroup is characteristic.