Modern Algebra 1, MATH 5410 Homework 6, Section I.8, Solutions Due Friday, October 23 at 1:40 Write in complete sentences!!! Explain what you are doing and convince me that you understand what you are doing and why. Justify all steps by quoting relevant results from the textbook or hypotheses. I.8.5. Let G and H be nite cyclic groups. Then G H is cyclic if and only if gcd(|G|, |H|) = (|G|, |H|) = 1. (This is Fraleigh's Theorem 11.5.) HINT: Use Exercises I.3.2 and I.3.6. I.8.9. If group G satises G = H K, where H = H {eK } < G and K = {eH } K < G, then H G/K and K G/H . HINT: Use projections and the First Isomorphism Theorem = = (Corollary I.5.7). I.8.12(a) A normal subgroup H of a group G is a direct factor (or \"direct summand\" under additive notation) if there exists a normal subgroup K of G such that G H K. Prove that if J is = a direct factor of K and K is a direct factor of G, then J is isomorphic to a normal subgroup of G. HINT: Use a projection and Theorem I.5.5. NOTE: We saw in Exercise I.5.10 that normality is not transitive; that is, A B and B C does not imply A C. However, normality combined with the property of being a direct factor does imply normality (up to isomorphism). I.8.14. Let H1 G1 and H2 G2 . Give examples to show that each of the following statements is false. HINT: (a), (c), and (d) can be done with cyclic groups. For (b), consider Dn where n is even. (a) G1 G2 and H1 H2 implies that G1 /H1 G2 /H2 . = = = (b) G1 G2 and G1 /H1 G2 /H2 implies that H1 H2 . = = = (c) H1 H2 and G1 /H1 G2 /H2 implies that G1 G2 . = = = (d) (G1 /H1 ) H1 G1 . = I.8.A. (Bonus) With the notation of Theorem I.8.5, verify that is a homomorphism and i = i for all i