Difference between revisions of "Group"
From Maths
(Created page with "==Definition== A group is a set {{M|G}} and an operation <math>*:G\times G\rightarrow G</math>, denoted <math>(G,*:G\times G\rightarrow G)</math> but Mathematicians are lazy...") |
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| All elements of {{M|G}} have an [[Inverse element|inverse element]] under {{M|*}}, that is | | All elements of {{M|G}} have an [[Inverse element|inverse element]] under {{M|*}}, that is | ||
|} | |} | ||
| + | ==Important theorems== | ||
| + | ===Identity is unique=== | ||
| + | {{Begin Theorem}} | ||
| + | Proof: | ||
| + | {{Begin Proof}} | ||
| + | Assume there are two identity elements, {{M|e}} and {{M|e`}} 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. | ||
| + | {{End Proof}} | ||
| + | {{End Theorem}} | ||
Revision as of 10:01, 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.
