Editing Group (mathematics) (section)

=== First example: the integers ===
One of the more familiar groups is the set of [[integer]]s 
<math display=block>\Z = \{\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots\}</math>
together with [[addition]].{{sfn|Lang|2005|loc=App. 2|p=360}} For any two integers <math>a</math> and {{tmath|1= b }}, the [[Summation|sum]] <math>a+b</math> is also an integer; this ''[[Closure (mathematics)|closure]]'' property says that <math>+</math> is a [[binary operation]] on {{tmath|1= \Z }}. The following properties of integer addition serve as a model for the group axioms in the definition below.
* For all integers {{tmath|1= a }}, <math>b</math> and&nbsp;{{tmath|1= c }}, one has {{tmath|1= (a+b)+c=a+(b+c) }}. Expressed in words, adding <math>a</math> to <math>b</math> first, and then adding the result to <math>c</math> gives the same final result as adding <math>a</math> to the sum of <math>b</math> and&nbsp;{{tmath|1= c }}. This property is known as ''[[associativity]]''.
* If <math>a</math> is any integer, then <math>0+a=a</math> and {{tmath|1= a+0=a }}. [[Zero]] is called the ''[[identity element]]'' of addition because adding it to any integer returns the same integer.
* For every integer {{tmath|1= a }}, there is an integer <math>b</math> such that <math>a+b=0</math> and {{tmath|1= b+a=0 }}. The integer <math>b</math> is called the ''[[inverse element]]'' of the integer <math>a</math> and is denoted&nbsp;{{tmath|1= -a }}.

The integers, together with the operation {{tmath|1= + }}, 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.