## 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)