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14. Find two groups G and H such that G * H, but Aut(G) ~ Aut(H). 15. Prove the assertion in Example 14 that the inner automorphisms PRO' PROO' DH, and d, of DA are distinct. 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 d(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 p be an automor- phism of D,. Determine o(R180). 24. Let o be an automorphism of a group G. Prove that H = (x E GI 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 Z12 and a subgroup isomorphic to Zoo. No need to prove anything, but explain your reasoning. 26. Suppose that d: Z20 - Z20 is an automorphism and 4(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.)