This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups
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.
|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.|
Combining quasicyclic groups for all primes
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 -adics are constructed as an inverse limit for surjective maps , the quasicyclic group is constructed as a direct limit for injective maps .