Difference between revisions of "Group"
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(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 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:
