18. 19. 20. 21. 22. 23. 24. 25 26 27 29. 30. 31. 32. 33. 34. 35. Suppose that a is a group element and a6 E e. What are the possi- bilities for lal? Provide reasons for your answer. If a is a group element and a has innite order, prove that a'" 514 a" when m =75 1:. For any group elements a and b, prove that Iabl : Ibal. Show that if a is an element of a group G, then ta! 5 IGI. Show that U(l4) = (3) = (5). [Hence, U(l4) is cyclic.] Is U04) = (11)? Show that U00) a (k) for any k in 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 Dm either every member of the subgroup is a rotation or exactly half of the members are rotations. Let H be a subgroup of D\" of odd order. Prove that every member of H is a rotation. Prove that a group with two elements of order 2 that commute must have a subgroup of order 4. For every even integer it, show that ii) 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 2' under addition and that H con tains 250 and 350. What are the possibilities for H? Prove that the dihedral group of order 6 does not have a subgroup of order 4. HH 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).]