Difference between revisions of "Group"
From Maths
m |
m (→Identity is unique) |
||
| Line 27: | Line 27: | ||
That is both: | 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> | ||
| − | * <math>\forall g\in G[e | + | * <math>\forall g\in G[e'*g=g*e'=g]</math> |
| − | But then <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}} | ||
Revision as of 10:03, 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.
