Group

Definition

A group is a set [ilmath]G[/ilmath] and an operation [math]*:G\times G\rightarrow G[/math], denoted [math](G,*:G\times G\rightarrow G)[/math] but mathematicians are lazy so we just write [math](G,*)[/math]

Such that the following axioms hold:

Axioms

Words	Formal
[math]\forall a,b,c\in G:[(a*b)*c=a*(b*c)][/math]	[ilmath]*[/ilmath] is associative, because of this we may write [math]a*b*c[/math] unambiguously.
[math]\exists e\in G\forall g\in G[e*g=g*e=g][/math]	[ilmath]*[/ilmath] has an identity element
[math]\forall g\in G\exists x\in G[xg=gx=e][/math]	All elements of [ilmath]G[/ilmath] have an inverse element under [ilmath]*[/ilmath], that is

Important theorems

Identity is unique

Proof:

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Assume there are two identity elements, [ilmath]e[/ilmath] and [ilmath]e`[/ilmath] with [math]e\ne e`[/math].

That is both:

• [math]\forall g\in G[e*g=g*e=g][/math]
• [math]\forall g\in G[e'*g=g*e'=g][/math]

But then [math]ee'=e[/math] and also [math]ee`=e'[/math] thus we see [math]e'=e[/math] contradicting that they were different.

Now we know the identity is unique, so we can give it a symbol:

Group	Identity element
[ilmath](G,+)[/ilmath] - additive notation [ilmath]a+b[/ilmath]	We denote the identity [ilmath]0[/ilmath], so [math]a+0=0+a=a[/math]
[ilmath](G,*)[/ilmath] - multiplicative notation [ilmath]ab[/ilmath]	We denote the identity [ilmath]1[/ilmath], so [math]1a=a*1=a[/math]
[ilmath]\text{GL}(n,F)[/ilmath] - the General linear group (All [ilmath]n\times n[/ilmath] matrices of non-zero determinant)	We denote the identity by [ilmath]Id,I,I_n[/ilmath] or sometimes [ilmath]Id_n[/ilmath] that is [math]AI=IA=A[/math]