h7 - 1 Homework VII: July 17, 2009 2 . 68 . Prove that a...

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Unformatted text preview: 1 Homework VII: July 17, 2009 2 . 68 . Prove that a group G is abelian if and only if the function f : G G , given by f ( a ) = a- 1 , is a homomorphism. Assume that G is abelian. If a,b G , then ( ab )- 1 = b- 1 a- 1 = a- 1 b- 1 ; that is, f ( ab ) = f ( a ) f ( b ). Conversely, assume that f is a homomorphism; that is, ( ab )- 1 = a- 1 b- 1 . Then aba- 1 b- 1 = ab ( ab )- 1 = 1, so that ab = ba , and G is abelian. 2 . 78 . Let G be the additive group of all polynomials in x with coefficients in Z , and let H be the multiplicative group of all positive rationals. Prove that G = H . List the prime numbers in ascending order: p = 2, p 1 = 3, p 2 = 5, etc., and define ( e + e 1 x + e 2 x 2 + + e n x n ) = p e p e 1 1 p e 2 2 p e n n . It is routine to check that is a homomorphism that is surjective. That is injective is the Fundamental Theorem of Arithmetic....
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This note was uploaded on 11/15/2010 for the course MATH 417 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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