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Unformatted text preview: Homework 5 for MATH 435 Solutions Problem 1 (a) Show that ( Q , +) is not cyclic. Solution. For any q = a/b ∈ Q , a , b ∈ Z relatively prime, < q > = { ma/b : m ∈ Z } . Then 1 / ( b + 1 ) < < q > , hence Q is not cyclic. (b) Does there exist finite X ⊆ Q such that < X > = Q (as an additive group)? Solution. Let q 1 = a 1 /b 1 , . . . , q n = a n /b n ∈ Q , a i , b i ∈ Z relatively prime. Then < q 1 , . . . , q n > ⊆ { m/ ( b 1 ··· b n ): m ∈ Z } , and hence 1 / ( b 1 ··· b n + 1 ) < < q 1 , . . . , q n > . Problem 2 Let G be a group and H a subgroup. Let x ∈ G . Let xHx 1 be the subset of G consisting of all elements xyx 1 , y ∈ H . Show that xHx 1 is a subgroup of G Solution. xHx 1 inherits associativity from G . Since H is a subgroup, 1 ∈ H , and hence 1 = x 1 x 1 ∈ H . Finally, given y ∈ xHx 1 , again since H is a subgroup, y 1 ∈ H , and hence xy 1 x 1 ∈ H . xy 1 x 1 is the inverse of xyx 1 , because ( xyx 1 )( xy 1 x 1 ) = xyx 1 xy...
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 '08
 YEE
 Math, Group Theory, Equivalence relation, Cyclic group, Coset, Lagrange's theorem

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