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Homework5

# Homework5 - G Explain 4 Consider the permutation...

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Homework 5 Math 332, Spring 2010 These problems must be written up in L A T E X, and are due this Thursday, March 11. 1. Let G be the following subgroup of GL (2 , R ): G = ±² a - b b a ³ : a,b R and ( a,b ) 6 = (0 , 0) ´ . Prove that G is isomorphic to C # , the group of nonzero complex numbers under multiplication. 2. Let Aﬀ ( R ) be the group of all functions f : R R of the form f ( x ) = ax + b where a,b R and a 6 = 0. Prove that Aﬀ ( R ) is isomorphic to a subgroup of GL (2 , R ). 3. Let G = U (15), the multiplicative group of units modulo 15. (a) List the eight automorphisms of G . Write your answers as permutations of the set { 1 , 2 , 4 , 7 , 8 , 11 , 13 , 14 } . (b) What is the isomorphism type of Aut(
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Unformatted text preview: G )? Explain. 4. Consider the permutation representation of U (15) for which 2 7→ (1 2 3 4) and 14 7→ (5 6). (a) Find the image of each element of U (15) under this representation. (b) Find a matrix representation of U (15) using 6 × 6 permutation matrices. (c) Consider the following six vectors in R 3 : v 1 = (1 , , 0) v 2 = (0 , 1 , 0) v 3 = (-1 , , 0) v 4 = (0 ,-1 , 0) v 5 = (0 , , 1) v 6 = (0 , ,-1) . Use these vectors to ﬁnd a 3 × 3 matrix representation of U (15)....
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