Group of integers

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

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