Difference between revisions of "Group"
From Maths
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| + | Now we know the identity is unique, so we can give it a symbol: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Group | ||
| + | ! Identity element | ||
| + | |- | ||
| + | | {{M|(G,+)}} - additive notation {{M|a+b}} | ||
| + | | We denote the identity {{M|0}}, so <math>a+0=0+a=a</math> | ||
| + | |- | ||
| + | | {{M|(G,*)}} - multiplicative notation {{M|ab}} | ||
| + | | We denote the identity {{M|1}}, so <math>1a=a*1=a</math> | ||
| + | |- | ||
| + | | {{M|\text{GL}(n,F)}} - the [[General linear group]]<br/> | ||
| + | (All {{M|n\times n}} matrices of non-zero [[Determinant|determinant]]) | ||
| + | | We denote the identity by {{M|Id,I,I_n}} or sometimes {{M|Id_n}}<br/>that is <math>AI=IA=A</math> | ||
| + | |} | ||
Revision as of 10:15, 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:
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 |
| \text{GL}(n,F) - the General linear group (All n\times n matrices of non-zero determinant) |
We denote the identity by Id,I,I_n or sometimes Id_n that is AI=IA=A |
