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 G and an operation ∗:G×G→G, denoted (G,∗:G×G→G) but mathematicians are lazy so we just write (G,∗)
Such that the following axioms hold:
Axioms
| Words | Formal |
|---|---|
| ∀a,b,c∈G:[(a∗b)∗c=a∗(b∗c)] | ∗ is associative, because of this we may write a∗b∗c unambiguously. |
| ∃e∈G∀g∈G[e∗g=g∗e=g] | ∗ has an identity element |
| ∀g∈G∃x∈G[xg=gx=e] | All elements of G have an inverse element under ∗, that is |
Important theorems
Identity is unique
[Expand]
Proof:
