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
[hide]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 that the identity is unique. (Method: assume e and e′ with e≠e′ are both identities, reach a contradiction)
Now we know the identity is unique, so we can give it a symbol:
| Group | Identity element |
|---|---|
| (G,+) - additive notation a+b | We denote the identity 0, so a+0=0+a=a |
| (G,∗) - multiplicative notation ab | We denote the identity 1, so 1a=a∗1=a |
| GL(n,F) - the General linear group (All n×n matrices of non-zero determinant) |
We denote the identity by Id,I,In or sometimes Idn that is AI=IA=A |
Inverse is unique
[Expand]
Proof that the inverse is unique. (Suppose that x and x′ are both inverses with x≠x′ and reach a contradiction
Cancellation laws
These are extremely important.
- ab=ac⟹b=c
- ba=ca⟹b=c
[Expand]
Proof
