19. 20. 21. 22. 23. 24. 25. 26. 27 29. 30. 31. 32. 33. 34. 35. d If a is a group element and a has innite order, prove that a'" as a" when m a n. For any group elements a and b, prove that lab! = lbal. Show that if a is an element of a group G, then Ial E IGI. Show that U(l4) = (3) = (5). [Hence, U(l4) is cyclic.] Is U(l4) = (l 1)? Show that U(20) 95 (k) for any kin U(20). [Hence, U(20) is not cyclic.] Suppose n is an even positive integer and H is a subgroup of Zn. Prove that either every member of H is even or exactly half of the members of H are even. Let n be a positive even integer and let H be a subgroup of Zn of odd order. Prove that every member of H is an even integer. Prove that for every subgroup of D, either every member of the subgroup IS a rotation or exactly half of the members are e:m'tionS-f . LetH be a subgroup of D of 0d order. Prove H is a rotation. . Prove that a group with two 1e gm must have a subgroup of order 4. For every even integer :1, show that D has a subgroup of order 4. Suppose that H is a proper subgroup of Z under addition and H con- tains 18, 30, and 40. Determine H. Suppose that H is a proper subgroup of Z under addition and that H contains 12, 30, and 54. What are the possibilities for H? Suppose that H is a subgroup of Z under addition and that H con- tains 25 and 350. What are the possibilities for H? Prove that the dihedral group of order 6 does not have a subgroup of order 4. IfH and K are subgroups of G, show that H ['1 K is a subgroup of G. (Can you see that the same proof shows that the intersection of any number of subgroups of G, nite or innite, is again a subgroup of G?) Let G be a group. Show that Z(G) = aEGCQa). [This means the intersection of all subgroups of the form C(a).]