Group of integers

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

GAP implementation
The group can be defined using the FreeGroup function:

FreeGroup(1)