Group of integers
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Contents
Definition
Usual definition
The group of integers, typically denoted , is defined as follows:
- The underlying set is the set of all integers
- The group operation is integer addition
- The identity element is the integer
- The inverse map is the additive inverse, sending an integer
to the integer
In the 4-tuple notation, the group of integers in the group .
Other definitions
Some other equivalent formulations of the group of integers:
- It is the additive group of the ring of integers
- It is the infinite cyclic group
- It is the free group on one generator
- It is the free abelian group on one generator
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order ((number of elements, equivalently, cardinality or size of underlying set) | Infinite (countable) | Not a finite group. |
exponent | Infinite | Not a periodic group. |
derived length | 1 | The group is an abelian group. |
nilpotency class | 1 | The group is an abelian group. |
Fitting length | 1 | The group is an abelian group. |
Frattini length | 1 | The group is a Frattini-free group. |
subgroup rank of a group | 1 | The group is cyclic, hence so is every subgroup. |
Group properties
Property | Satisfied | Explanation | Comment |
---|---|---|---|
cyclic group | Yes | ||
abelian group | Yes | Cyclic implies abelian | |
finite group | No | ||
finitely generated group | Yes | Generating set of size one. | |
slender group | Yes | Every subgroup is cyclic. | |
Hopfian group | Yes | Not isomorphic to any proper quotient, which is finite. | |
co-Hopfian group | No | Isomorphic to the proper subgroup generated by any element not the generator or the identity. |
GAP implementation
The group can be defined using the FreeGroup function:
FreeGroup(1)