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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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