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Definition
Let [ilmath]G[/ilmath] be a set and a binary operation (a function) [ilmath]*:G\times G\rightarrow G[/ilmath], with the following properties[1]:
- [ilmath]\forall g,h,k\in G[(g*h)*k=g*(h*k)][/ilmath] - is associative
- [ilmath]\exists e\in G\ \forall g\in G[g*e=e*g=g][/ilmath] - there exists an identity element of [ilmath]G[/ilmath][Note 1]
- [ilmath]\forall g\in G\exists h\in G[g*h=h*g=e][/ilmath] - for each element there exists an inverse element in [ilmath]G[/ilmath][Note 2]
If [ilmath]*[/ilmath] satisfies these 3 properties than the tuple, [ilmath](G,*:G\times G\rightarrow G)[/ilmath] - or just [ilmath](G,*)[/ilmath] as mathematicians are lazy, is called a group.
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 [ilmath](G,*)[/ilmath] satisfies and additional property:
- [ilmath]\forall g,h\in G[g*h=h*g][/ilmath] - the operation is commutative
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
- ↑ At this point we do not know that the identity element is unique, there could be more than one such [ilmath]e[/ilmath] - but one exists. In fact it is unique, as we will see later
- ↑ Again, we do not know there is a unique inverse, or for which of the [ilmath]e[/ilmath] elements the equality refers to (if there are even more than one).
- There is actually only one identity element, only one [ilmath]e\in G[/ilmath] and also only one inverse for each element, but this requires proof.
References
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