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# h7 - 1 Homework VII 2.68 Prove that a group G is abelian if...

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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 0 = 2, p 1 = 3, p 2 = 5, etc., and define ϕ ( e 0 + e 1 x + e 2 x 2 + · · · + e n x n ) = p e 0 0 p e 1 1 p e 2 2 · · · p e n n . It is routine to check that ϕ is a homomorphism that is surjective. That ϕ is injective is the Fundamental Theorem of Arithmetic. 2 . 81 . Define W = (1 2)(3 4) , the cyclic subgroup of S 4 generated by (1 2)(3 4). Show that W V , where V is the four-group, and that V S 4 , but that W is not a normal subgroup of S 4 . W is a normal subgroup of V , for V is abelian, and every subgroup of an abelian group is normal. The four-group V is a normal subgroup of S 4 : Proposition 2.33 says that two permutations are conjugate in S n if and only if they have the same cycle structure. But there are only three permutations in
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