Chapter4 - Chapter 4 Group Actions Keith E Emmert Group...

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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 Conjugation-The 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 Conjugation-The Class Equation Automorphisms Sylow’s Theorem Overview Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication-Cayleys The Groups Acting on Themselves by Conjugation-The 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 Conjugation-The 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 Conjugation-The Class Equation Automorphisms Sylow’s Theorem Overview Group Actions and Permutation Representations Groups Acting on Themselves by Left Multiplication-Cayleys The Groups Acting on Themselves by Conjugation-The 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 Conjugation-The 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 Conjugation-The 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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Chapter4 - Chapter 4 Group Actions Keith E Emmert Group...

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