Quasicyclic group
From Groupprops
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
Contents
Definition
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.
Particular cases
Prime number ![]() |
![]() |
---|---|
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 -quasicyclic groups for all prime numbers
is isomorphic to
, 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 -adics are constructed as an inverse limit for surjective maps
, the quasicyclic group is constructed as a direct limit for injective maps
.