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== Definition (left-sided) == | == Definition (left-sided) == | ||
| − | A group <math>\langle G, \cdot \rangle</math> is a set G and a [[Binary Operation_OldKiwi]] <math>\cdot</math> on G such that the group axioms hold: | + | A group <math>\langle G, \cdot \rangle</math> is a set G and a [[Binary Operation_OldKiwi]] <math>\cdot</math> on G (closed over G by definition) such that the group axioms hold: |
#Associativity: <math>a\cdot(b\cdot c) = (a\cdot b)\cdot c</math> <math>\forall a,b,c \in G</math> | #Associativity: <math>a\cdot(b\cdot c) = (a\cdot b)\cdot c</math> <math>\forall a,b,c \in G</math> | ||
#Identity: <math>\exists e\in G</math> such that <math>e\cdot a = a</math> <math>\forall a \in G</math> | #Identity: <math>\exists e\in G</math> such that <math>e\cdot a = a</math> <math>\forall a \in G</math> | ||
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=== Element commutes with inverse === | === Element commutes with inverse === | ||
<math>Thm: \forall a\in G</math> <math>a\cdot a^{-1} = a^{-1}\cdot a = 1</math> | <math>Thm: \forall a\in G</math> <math>a\cdot a^{-1} = a^{-1}\cdot a = 1</math> | ||
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| + | <math>Prf: Since a^{-1}\in G, its inverse (a^{-1})^{-1}\in G with (a^{-1})^{-1}\cdot a^{-1} = 1 by the inverse axiom. But also a^{-1}\cdot a = 1</math> | ||
Revision as of 22:46, 13 May 2008
Definition (left-sided)
A group $ \langle G, \cdot \rangle $ is a set G and a Binary Operation_OldKiwi $ \cdot $ on G (closed over G by definition) such that the group axioms hold:
- Associativity: $ a\cdot(b\cdot c) = (a\cdot b)\cdot c $ $ \forall a,b,c \in G $
- Identity: $ \exists e\in G $ such that $ e\cdot a = a $ $ \forall a \in G $
- Inverse: $ \forall a\in G $ $ \exists a^{-1}\in G $ such that $ a^{-1}\cdot a = e $
Notation
Groups written additively use + to denote their Binary Operation_OldKiwi, 0 to denote their identity, $ -a $ to denote the inverse of element $ a $, and $ na $ to denote $ a + a + \ldots + a $ ($ n $ terms).
Groups written multiplicatively use $ \cdot $ or juxtaposition to denote their Binary Operation_OldKiwi, 1 to denote their identity, $ a^{-1} $ to denote the inverse of element $ a $, and $ a^n $ to denote $ a \cdot a \cdot \ldots \cdot a $ ($ n $ terms).
Theorems
Element commutes with inverse
$ Thm: \forall a\in G $ $ a\cdot a^{-1} = a^{-1}\cdot a = 1 $
$ Prf: Since a^{-1}\in G, its inverse (a^{-1})^{-1}\in G with (a^{-1})^{-1}\cdot a^{-1} = 1 by the inverse axiom. But also a^{-1}\cdot a = 1 $
