Difference between revisions of "Group"
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| + | Now we know the identity is unique, so we can give it a symbol: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Group | ||
| + | ! Identity element | ||
| + | |- | ||
| + | | {{M|(G,+)}} - additive notation {{M|a+b}} | ||
| + | | We denote the identity {{M|0}}, so <math>a+0=0+a=a</math> | ||
| + | |- | ||
| + | | {{M|(G,*)}} - multiplicative notation {{M|ab}} | ||
| + | | We denote the identity {{M|1}}, so <math>1a=a*1=a</math> | ||
| + | |- | ||
| + | | {{M|\text{GL}(n,F)}} - the [[General linear group]]<br/> | ||
| + | (All {{M|n\times n}} matrices of non-zero [[Determinant|determinant]]) | ||
| + | | We denote the identity by {{M|Id,I,I_n}} or sometimes {{M|Id_n}}<br/>that is <math>AI=IA=A</math> | ||
| + | |} | ||
Revision as of 10:15, 11 March 2015
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:
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] |
