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332f10hw8 - faithful i.e the kernel g ∈ G g x = x is...

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Math 332 – Abstract Algebra Assignment 8 – Due November 11, 2010 1. Do problems 6, 7, 15 in chapter 11. 2. Let G be a group acting on itself by conjugation (inner automorphisms) so that the group action is given explicitly as: G |{z} group × G |{z} set -→ G |{z} set ( g, h ) 7→ ghg - 1 . (a) Prove or give a counterexample: this action is
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Unformatted text preview: faithful ( i.e. the kernel { g ∈ G : g · x = x } is trivial). (b) Explain why this action is never transitive if | G | > 1. (c) Determine all groups G for which this action is trivial. 3. Let G be a group acting on a set X . Show that for any x ∈ X , we have G gx = gG x g-1 ....
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