structural:scholar_edition, linguistic:voice_change
First example: the integers

One of the more familiar groups is the set of integers Z = { …, − 4, − 3, − 2, − 1, 0, 1, 2, 3, 4, … } {\displaystyle \mathbb {Z} =\{\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots \}} together with addition. For any two integers a {\displaystyle a} and b {\displaystyle b}, the sum a + b {\displaystyle a+b} is also an integer; this closure property says that + {\displaystyle +} is a binary operation on Z {\displaystyle \mathbb {Z} }. The following properties of integer addition serve as a model for the group axioms in the definition below.

The integers, together with the operation + {\displaystyle +}, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.