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Unformatted text preview: a,b A and b = g.a for some g G , then G b = gG a g1 ( G a is the stabilizer of a ). Deduce that if G acts transitively on A then the kernel of the action is g G gG a g1 . (b) Let G be a permutation group on the set A (i.e., G < S A ), let G and let a A . Prove that G a 1 = G ( a ) . Deduce that if G acts transitively on A then G G a 1 = { e } . (c) Assume that G is an abelian, transitive subgroup of S A . Show that ( a ) = a for all G { e } and for all a A . Deduce that  G  =  A  . (Hint: use the preceding exercise.) 7. Section 4.2, Exercise 4: Use the left regular representation of the quaternionic group Q 8 to produce two elements of S 8 which generate a subgroup of S 8 isomorphic to Q 8 . 1...
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 Fall '08
 OGUS
 Math, Algebra

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