Difference between revisions of "Group"
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===Identity is unique=== | ===Identity is unique=== | ||
{{Begin Theorem}} | {{Begin Theorem}} | ||
| − | Proof: | + | Proof that the identity is unique. (Method: assume {{M|e}} and {{M|e'}} with <math>e\ne e'</math> are both identities, reach a contradiction) |
{{Begin Proof}} | {{Begin Proof}} | ||
| − | Assume there are two identity elements, {{M|e}} and {{M|e | + | Assume there are two identity elements, {{M|e}} and {{M|e'}} with <math>e\ne e'</math>. |
That is both: | That is both: | ||
| Line 29: | Line 29: | ||
* <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 | + | But then <math>ee'=e</math> and also <math>ee'=e'</math> thus we see <math>e'=e</math> contradicting that they were different. |
{{End Proof}} | {{End Proof}} | ||
{{End Theorem}} | {{End Theorem}} | ||
| Line 49: | Line 49: | ||
| We denote the identity by {{M|Id,I,I_n}} or sometimes {{M|Id_n}}<br/>that is <math>AI=IA=A</math> | | We denote the identity by {{M|Id,I,I_n}} or sometimes {{M|Id_n}}<br/>that is <math>AI=IA=A</math> | ||
|} | |} | ||
| + | |||
| + | ===Inverse is unique=== | ||
| + | {{Begin Theorem}} | ||
| + | Proof that the inverse is unique. (Suppose that <math>x</math> and {{M|x'}} are both inverses with {{M|x\ne x'}} and reach a contradiction | ||
| + | {{Begin Proof}} | ||
| + | {{Todo}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
| + | |||
| + | ===Cancellation laws=== | ||
| + | These are extremely important. | ||
| + | # <math>ab=ac\implies b=c</math> | ||
| + | # <math>ba=ca\implies b=c</math> | ||
| + | {{Begin Theorem}} | ||
| + | Proof | ||
| + | {{Begin Proof}} | ||
| + | {{Todo}} | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
Revision as of 12:37, 11 March 2015
Contents
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 that the identity is unique. (Method: assume [ilmath]e[/ilmath] and [ilmath]e'[/ilmath] with [math]e\ne e'[/math] are both identities, reach a contradiction)
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] |
Inverse is unique
Proof that the inverse is unique. (Suppose that [math]x[/math] and [ilmath]x'[/ilmath] are both inverses with [ilmath]x\ne x'[/ilmath] and reach a contradiction
TODO:
Cancellation laws
These are extremely important.
- [math]ab=ac\implies b=c[/math]
- [math]ba=ca\implies b=c[/math]
Proof
TODO:
