# hw16 - H ≤ G Prove that G/H is abelian Exercise 8 Show...

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Modern Algebra 1 Homework 7.2 Spring 2010 Due March 10 Exercise 4. Let f : G H be a homomorphism of groups. a. If K C H prove that f - 1 ( K ) C G . b. If K C G and f is surjective prove that f ( K ) C H . Exercise 5. Let G be a group and H,K G . a. Prove that if H C G then HK = KH and HK G . b. Prove that if G is abelian then HK G . [ Suggestion: Use part a .] Exercise 6. To what familiar group is D n / h r i isomorphic? Be sure to justify your answer. Exercise 7. Let G be an abelian group and
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Unformatted text preview: H ≤ G . Prove that G/H is abelian. Exercise 8. Show that Q / Z is inﬁnite, but that every element in it has ﬁnite order. Exercise 9. Suppose that G = h a i is a cyclic group and H ≤ G . Prove that G/H is cyclic. What is it generated by? [ Suggestion: Use the canonical surjection f : G → G/H and exercise 5b from homework 4.]...
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## This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Spring '10 term at Trinity University.

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