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Unformatted text preview: Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on Themselves by ConjugationThe Class Equation Automorphisms Sylow’s Theorem Chapter 4: Group Actions Keith E. Emmert Tarleton State University March 5, 2010 Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on Themselves by ConjugationThe Class Equation Automorphisms Sylow’s Theorem Overview Group Actions and Permutation Representations Groups Acting on Themselves by Left MultiplicationCayleys The Groups Acting on Themselves by ConjugationThe Class Equatio Automorphisms Sylow’s Theorem Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on Themselves by ConjugationThe Class Equation Automorphisms Sylow’s Theorem Read This Section Yeah, read it! Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on Themselves by ConjugationThe Class Equation Automorphisms Sylow’s Theorem Overview Group Actions and Permutation Representations Groups Acting on Themselves by Left MultiplicationCayleys The Groups Acting on Themselves by ConjugationThe Class Equatio Automorphisms Sylow’s Theorem Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on Themselves by ConjugationThe Class Equation Automorphisms Sylow’s Theorem Transitive Definition 1 Let G be a group acting on the nonempty set A . 1. The equivalence class { g · a  g ∈ G } is called the orbit of G containing a . 2. The action of G on A is called transitive if there is only one orbit, i.e., given any two elements a , b ∈ A there is some g ∈ G such that a = g · b . Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on Themselves by ConjugationThe Class Equation Automorphisms Sylow’s Theorem Some Theory Theorem 2 Let G be a group, let H be a subgroup of G and let G act by left multiplication on the set A of left cosets of H in G. Let π H be the associated permutation representation afforded by this action. Then 1. G acts transitively on A. 2. the stabilizer in G of the point 1 H ∈ A is the subgroup H. 3. the kernel of the action (i.e., the kernel of π H ) is intersectiondisplay x ∈ G xHx 1 and Ker ( π H ) is the largest normal subgroup of G contained in H. Chapter 4: Group Actions Keith E. Emmert Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication Cayleys Theorem Groups Acting on...
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 Spring '08
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 Multiplication, Conjugacy class, group actions, Keith E. Emmert

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