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16. Find Aut(Z6). 17. If G is a group, prove that Aut(G) and Inn(G) are groups. (This exercise is referred to in this chapter.) 18. If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H). 19. Suppose d belongs to Aut(Z ) and a is relatively prime to n. If p(a) = b, determine a formula for b(x). 20. Let H be the subgroup of all rotations in D, and let o be an automor- phism of D . Prove that o( H) = H. (In words, an automorphism of D, carries rotations to rotations.) 21. Let H = (B E S, I B(1) = 1} and K = (B E S, | B(2) = 2). Prove that H is isomorphic to K. Is the same true if S is replaced by S,,, where n 2 3? 22. Show that Z has infinitely many subgroups isomorphic to Z. 23. Let n be an even integer greater than 2 and let o be an automor- phism of D. Determine d(R180). 24. Let o be an automorphism of a group G. Prove that H = (x E G | p(x) = x} is a subgroup of G. 25. Give an example of a cyclic group of smallest order that contains both a subgroup isomorphic to Z, and a subgroup isomorphic to Zzo. No need to prove anything, but explain your reasoning. 26. Suppose that d: Z20 - Z20 is an automorphism and o(5) = 5. What are the possibilities for d(x)? 27. Identify a group G that has subgroups isomorphic to Z, for all posi- tive integers n. 28. Prove that the mapping from U(16) to itself given by x -> x3 is an automorphism. 29. Let r E U(n). Prove that the mapping a: Z, -> Z, defined by a(s) = sr mod n for all s in Z, is an automorphism of Z. (This exercise is referred to in this chapter.) 30. The group a EZ is isomorphic to what familiar group? What if Z is replaced by R? 31. If and y are isomorphisms from the cyclic group (a) to some group and o(a) = y(a), prove that o = y. 32. Suppose that d: Z50 -> Zso is an automorphism with d(7) = 13