Group of integers

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

Usual definition

The group of integers, typically denoted $\mathbb {Z}$, 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 $0$
• The inverse map is the additive inverse, sending an integer $n$ to the integer $-n$

In the 4-tuple notation, the group of integers in the group $(\mathbb {Z} ,+,0,-)$.

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)

See also

• Multiplicative monoid of non-zero integers
• Additive group of complex numbers
• Additive group of real numbers
• Additive group of real algebraic numbers
• Additive group of rational numbers