Problem 4. We say that a group G is finitely generated if there is a finite subset S = {x1,...,xn} C G such that G = (S) = (x1, . . . Xn). 4.1. Let H be a finitely generated subgroup of the additive group Q of rational numbers. Prove that H is contained in cyclic group. 4.2. Using the above, prove that Q is not itself finitely generated. Hint. For 4.1, you may wish to proceed as follows: Since H is finitely generated, we can write H = (x1,...,xn) for some n > 0 and x1,...,xn Є Q. Write xi = ai/bi, and = let b = b₁b2bn. Show that H <≤

Problem 4. We say that a group G is finitely generated if there is a finite subset S = {x1,...,xn} C G such that G = (S) = (x1, . . . Xn). 4.1. Let H be a finitely generated subgroup of the additive group Q of rational numbers. Prove that H is contained in cyclic group. 4.2. Using the above, prove that Q is not itself finitely generated. Hint. For 4.1, you may wish to proceed as follows: Since H is finitely generated, we can write H = (x1,...,xn) for some n > 0 and x1,...,xn Є Q. Write xi = ai/bi, and = let b = b₁b2bn. Show that H <≤

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 27E

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