(New page: == Group Theory == Unfortunately, abstract algebra is not typically part of the ECE/CS curriculum. Here is a very brief overview/review of the group theoretic concepts involved in the 3-...) |
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3. Each element has an inverse: <math>\forall x\in G ~~ \exists x^{-1}\in G</math> s.t. <math>x\cdot x^{-1} = x^{-1}\cdot x = e</math> | 3. Each element has an inverse: <math>\forall x\in G ~~ \exists x^{-1}\in G</math> s.t. <math>x\cdot x^{-1} = x^{-1}\cdot x = e</math> | ||
| − | One particularly useful example is the [[EE662Sp10SymmetricGroup|Symmetric Group]] | + | One particularly useful example is the [[EE662Sp10SymmetricGroup|Symmetric Group]]. |
| + | An important concept in the 3-25-10 and 3-30-10 lectures is that | ||
| + | of a [[EE662Sp10GroupAction|Group Action]]. | ||
--[[User:Jvaught|Jvaught]] 20:38, 30 March 2010 (UTC) | --[[User:Jvaught|Jvaught]] 20:38, 30 March 2010 (UTC) | ||
Revision as of 17:17, 30 March 2010
Group Theory
Unfortunately, abstract algebra is not typically part of the ECE/CS curriculum. Here is a very brief overview/review of the group theoretic concepts involved in the 3-25-10 and 3-30-10 lectures.
A group is a set $ G $ along with a binary operation $ \cdot $ under which the set is closed such that the following group axioms hold.
1. The operation is associative 2. There is an identity element: $ \exists e\in G $ s.t. $ x\cdot e = e\cdot x = x ~~\forall x\in G $ 3. Each element has an inverse: $ \forall x\in G ~~ \exists x^{-1}\in G $ s.t. $ x\cdot x^{-1} = x^{-1}\cdot x = e $
One particularly useful example is the Symmetric Group.
An important concept in the 3-25-10 and 3-30-10 lectures is that of a Group Action.
--Jvaught 20:38, 30 March 2010 (UTC)
