9Normal Subgroups and Factor Groups2019. Prove that if H has index 2 in G, then H is normal in G. (This exer-cise is referred to in Chapters 24 and 25 and this chapter.)ggctd10. Let H {(1), (12)(34)} in A,.a. Show that H is not normal in A,.nb. Referring to the multiplication table for A, in Table 5.1 on page111, show that, although a H = a,H and a,H = a,,H, it is notthat agasH= a,a1cosets of H do not form a group under coset multiplication.H.Explain why this proves that the left11. Let G Z U(4), H = ((2, 3), and K =is not isomorphic to G/K. (This shows that H~ K does not implythat G/H GIK.)12. Prove that a factor group of a cyclic group is cyclic.13. Prove that a factor group((2, 1)). Show that G/HofAbelian group is Abelian.14. What is the order of the element 14 (8) in the factorangroupZn(8)?15. What is the order of the element 4U(105) in the factor groupU(105)/U,(105)?16. Recall that Z(D) =RZ(D) in the factor group DdZ(D?{Rp, R). What is the order of the element18060ZI20) and H = (4)/(20). List the elements of H and G/H.17. Let G18. What is the order of the factor group Z15)?19. What is the order of the factor group (Z U(10))/(2, 9))?20. Construct the Cayley table for U(20)/U(20).601021. Prove that an Abelian group of order 33 is cyclic.22. Determine the order of (Z D Z)((2, 2). Is the group cyclic?Z)/((4, 2). Is the group cyclic?23. Determine the order of (Z24. The group (Z, Z2)(2, 2)) is isomorphic to one of Zg, Z Z, orZ, Z, Z,. Determine which one by elimination.25. Let G U(32) and H (1, 31). The group GIH is isomorphic toone of Zg, Z Z, or Z, e Z, Z. Determine which one byelimination.426. Let G be the group of quaternions given by the table in Exercise 4of the Supplementary Exercises for Chapters 1-4, and let H be thesubgroup {e, a2}. Is G/H isomorphic to Z, or Z, Z,?27. Let G = U(16), H = {1, 15), and K {1, 9}. Are H and K iso-morphic? Are G/H and G/K isomorphic?28. Let G Z, Z, H= {(0, 0), (2, 0), (0, 2), (2, 2)}, and K= {(1, 2)).Is GIH isomorphic to Z, or Z, Z,? Is GIK isomorphic to Z, orZ,Oz?4429. Prove that AOZ has no subgroup of order 18Naull