tutior const ০ 220 Groups 7. If is a homomorphism from G to H and o is a homomorphism from H to K, show that od is a homomorphism from G to K. How and Ker od related? If and o are onto and G is...

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Extracted text: tutior const ০ 220 Groups 7. If is a homomorphism from G to H and o is a homomorphism from H to K, show that od is a homomorphism from G to K. How and Ker od related? If and o are onto and G is finite, are Ker describe [Ker od:Ker ] in terms of IHI and IKI. 8. Let G be a group of permutations. For each or in G, define +1 if o is an even permutation, if o is an odd permutation. sgn(o) Prove that sgn is a homomorphism from G to the multiplicative group +1, -1}. What is the kernel? Why does this homomor- phism allow you index 2? Why does this prove Exercise 23 of Chapter 5? 9. Prove that the mapping from G H to G given by (g, h)-g is a homomorphism. What is the kernel? This mapping is called the to conclude that A, is a normal subgroup of S of n projection of G H onto G. 10. Let G be a subgroup of some dihedral group. For each x in G, define 202guo +1 if x is a rotation, -1 if x is a reflection. Ф(х) a homomorphism from G to the multiplicative Prove that is group {+1, -1}. What is the kernel? Why does this prove Exercise 25 of Chapter 3? 11. Prove that (Z Z)/(((a, 0)) X ((0, b))) 12. Suppose that k is a divisor of n. Prove that Z/(k) Z 13. Prove that (A B)/(A {e}) ~ B. 14. Explain why the correspondence x3x from Z2 to Z0 is not a homomorphism. 15. Suppose that is a homomorphism from Z0 to Za0 and Ker (0, 10, 20). If d(23) = 9, determine all elements that map to 9. 16. Prove that there is no homomorphism from Zg Z, onto Z4 Z 17. Prove that there is no homomorphism from Z,16 Z, onto ZgZ 18. Can there be a homomorphism from Z, Z onto Z? Can there be a homomorphism from Z16 onto Z, Z,? Explain your answers. 19. Suppose that there is a homomorphism from Z1, and that is not one-to-one. Determine . 20. How many homomorphisms are there from Z onto Z? How many are there to Z? 21. If is a homomorphism from Z onto a group of order 5, deter- mine the kernel of d. isomorphic to Z®Z. is 10 11 30 to some group 20 www