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==Definition==
Let {{M|G}} be a [[set]] and a [[binary operation]] (a [[function]]) {{M|*:G\times G\rightarrow G}}, with the following properties{{rFOAANHM}}:
# {{M|1=\forall g,h,k\in G[(g*h)*k=g*(h*k)]}} - is [[associative]]
# {{M|1=\exists e\in G\ \forall g\in G[g*e=e*g=g]}} - there exists an identity element of {{M|G}}<ref group="Note">At this point we do not know that the identity element is unique, there could be more than one such {{M|e}} - but one exists. In fact it is unique, as we will see later</ref>
# {{M|1=\forall g\in G\exists h\in G[g*h=h*g=e]}} - for each element there exists an inverse element in {{M|G}}<ref group="Note">Again, we do not know there is a unique inverse, or for which of the {{M|e}} elements the equality refers to (if there are even more than one).
* There is actually only one identity element, only one {{M|e\in G}} and also only one inverse for each element, but this requires proof.</ref>
If {{M|*}} satisfies these 3 properties than the [[tuple]], {{M|(G,*:G\times G\rightarrow G)}} - or just {{M|(G,*)}} as [[mathematicians are lazy]], is called a ''group''.<br/>
'''Claims: ''' (see below for proof)
* '''Claim 1: ''' The identity element is unique
* '''Claim 2: ''' The inverse element of an element is unique
===Abelian group===
If, additionally, a group {{M|(G,*)}} satisfies and additional property:
<ol start="4">
<li> {{M|1=\forall g,h\in G[g*h=h*g]}} - the operation is [[commutative]]</li>
</ol>
then we call the group an ''Abelian group'' or ''commutative group''
==Terminology and notations==

==Proof of claims==
{{:Group/Claim 1: The identity element of a group is unique}}
{{:Group/Claim 2: The inverse element for each element of a group is unique}}
==Notes==
<references group="Note"/>
==References==
<references/>
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{{Definition|Abstract Algebra|Group Theory}}