College Algebra Exam Review 227

College Algebra Exam Review 227 - gx for.g.x With this...

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Chapter 5 Actions of Groups 5.1. Group Actions on Sets We have observed that the symmetry group of the cube acts on various geometric sets associated with the cube, the set of vertices, the set of diag- onals, the set of edges, and the set of faces. In this section we look more closely at the concept of a group acting on a set. Definition 5.1.1. An action of a group G on a set X is a homomorphism from G into Sym .X/ . Let ' be an action of G on X . For each g 2 G , '.g/ is a bijection of X . The homomorphism property of ' means that for x 2 X and g 1 ;g 2 2 G , '.g 2 g 1 /.x/ D '.g 2 /.'.g 1 /.x// . Thus, if '.g 1 / sends x to x 0 , and '.g 2 / sends x 0 to x 00 , then '.g 2 g 1 / sends x to x 00 . When it cannot cause ambiguity, it is convenient to write
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Unformatted text preview: gx for '.g/.x/ . With this simplified notation, the homomorphism property reads like a mixed associative law: .g 2 g 1 /x D g 2 .g 1 x/ . Lemma 5.1.2. Given an action of G on X , define a relation on X by x ± y if there exists a g 2 G such that gx D y . This relation is an equivalence relation. Proof. Exercise 5.1.1 . n Definition 5.1.3. Given an action of G on X , the equivalence classes of the equivalence relation associated to the action are called the orbits of the action. The orbit of x will be denoted O .x/ . 237...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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