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Unformatted text preview: Homework 5 Solutions Math 332, Spring 2010 Problem 1. Proposition. Let G be the following subgroup of GL (2 , R ) : G = a- b b a : a,b R and ( a,b ) 6 = (0 , 0) . Then G is isomorphic to C # , the group of nonzero complex numbers under multiplication. Proof. Define : C # G by ( a + bi ) = a- b b a . Clearly is a bijection. Furthermore, if a + bi,c + di C # , then ( ( a + bi )( c + di ) ) = ( ( ac- bd ) + ( ad + bc ) i ) = ac- bd- ad- bc ad + bc ac- bd = a- b b a c- d d c = ( a + bi ) ( c + di ) , and therefore is an isomorphism. Problem 2. Proposition. 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 . Then Aff ( R ) is isomorphic to the group G = a b 0 1 : a,b R and a 6 = 0 . Proof. Let : Aff ( R ) G be the function that maps f ( x ) = ax + b to the matrix a b 0 1 . Clearly is a bijection. To prove that is an isomorphism, let f,g Aff ( R ), where f ( x ) = ax + b and...
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