Quiz 2 Practice Problems
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1 Quiz 2 Practice Problems Math 332, Spring 2010 Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and define a function ϕ: C C by ϕ(a + bi) = a bi. Prove that ϕ is an automorphism of C. 2. Let G be an abelian group, and define a function ϕ: G G by ϕ(a) = a 1. Prove that ϕ is an automorphism of G. 3. Let ϕ: Z 10 U(11) be an isomorphism, and suppose that ϕ(1) = 8. Determine ϕ(3). 4. Prove that A 4 is not isomorphic to D List the automorphisms of Z 8. Express your answers as permutations of the set {0, 1,..., 7}. 6. List the four elements of Inn(D 4 ). Express your answers as permutations of the set {e, r, r 2, r 3, s, rs, r 2 s, r 3 s}. 7. Let α Aut(D 5 ), and suppose that α(r) = r 2 and α(s) = rs. Find α(r 3 s). 8. Let α Aut(Q 8 ), and suppose that α(i) = k and α(j) = i. Express α as a permutation of the set {1, 1, i, i, j, j, k, k}. 9. Determine the isomorphism type of the group whose Cayley table is shown below: e p q r s t u v e e p q r s t u v p p r u t q e v s q q s r v t u p e r r t v e u p s q s s v p u r q e t t t e s p v r q u u u q t s e v r p v v u e q p s t r 1
2 Direct Products 10. Let G be a group, and define a function ϕ: G G G G G G by ϕ(a, b, c) = (b, c, a). Prove that ϕ is an automorphism of G G G. 11. Let G be a group, and let be the following subset of G G: Prove that is a subgroup of G G. = {(g, g) : g G}. 12. Let α be an automorphism of Z 5 Z 5 satisfying α(1, 4) = (2, 1) and α(0, 1) = (1, 0). Determine α(2, 0). 13. Find an element of order 12 in Q 8 S Find an element of order 6 in A 4 Z Determine the number of elements of order 6 in D 4 S Determine the number of elements of order 9 in Z 9 Z Determine the isomorphism type of each of the following groups. In each case, express your answer as a direct product of cyclic groups: (a) U(77) (b) U(165) (c) U(135) (d) U(72) 18. Determine the isomorphism type of Aut(Z 45 ). Express your answer as a direct product of cyclic groups. 19. Determine the isomorphism type of Aut ( U(25) ). Express your answer as a direct product of cyclic groups. 20. What is the largest order of any element of U(900)? 21. List four non-isomorphic groups of order List four non-isomorphic groups of order 30. 2
3 Cosets and Lagrange s Theorem 23. Let H = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. (a) Find the left cosets of H in A 4. (b) How many left cosets does H have in S 4? 24. Determine the left cosets of the subgroup {1, 11} in U(30). 25. Determine the left cosets of the subgroup {(0, 0, 0), (1, 1, 1)} in Z 2 Z 2 Z Let G be a group of order pq, where p and q are prime. Prove that every proper subgroup of G is cyclic. 27. Compute the following: (a) 5 15 mod 7 (b) 7 13 mod 11 (c) 8 50 mod Suppose that a group contains elements of orders 1 through 10. What is the minimum possible order of the group? 29. Let G be the permutation group with the following eight elements: { e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 3)(5 6), (2 4)(5 6), ( )(5 6), ( )(5 6) } (a) Find the stabilizer of 1 in G. (b) Find the orbit of 1 under G. (c) Find the stabilizer in 5 in G. (d) Find the orbit of 5 under G. 30. Let G be a permutation group of order 60, and suppose that the orbit of 1 under G is {1, 3, 4, 6}. Determine the order of the stabilizer of 1 in G. 3
4 Answers 1. Clearly ϕ is a bijection. Furthermore, if a + bi, c + di C, then ϕ ( (a + bi) + (c + di) ) = ϕ ( (a + c) + (b + d)i ) = (a + c) (b + d)i = (a bi) + (c di) = ϕ(a + bi) + ϕ(c + di), 2. Clearly ϕ is a bijection. Furthermore, if a, b G, then ϕ(ab) = (ab) 1 = b 1 a 1 = a 1 b 1 = ϕ(a)ϕ(b), 3. ϕ(3) = 8 3 mod 11 = D 6 has an element of order 6, but A 4 does not. 5. e, (1 3)(2 6)(5 7), (1 5)(3 7), (1 7)(2 6)(3 5) 6. e, (s r 2 s)(rs r 3 s), (r r 3 )(rs r 3 s), (r r 3 )(s r 2 s) 7. α(r 3 s) = α(r) 3 α(s) = (r 2 ) 3 (rs) = r 2 s. 8. (i k j)( i k j) 9. Q Clearly ϕ is a bijection. Furthermore, if (a, b, c), (a, b, c ) G G G, then ϕ ( (a, b, c)(a, b, c ) ) = ϕ(aa, bb, cc ) = (bb, cc, aa ) = (b, c, a)(b, c, a ) = ϕ(a, b, c)ϕ(a, b, c ), 11. We shall use the one-step subgroup test. Clearly is a nonempty subset of G G. Furthermore, if (g, g), (h, h), then (g, g)(h, h) 1 = (g, g)(h 1, h 1 ) = (gh 1, gh 1 ) which proves that is a subgroup of G G. 12. Since α(1, 0) = α(1, 4) + α(0, 1) = (2, 1) + (1, 0) = (3, 1), we conclude that α(2, 0) = α(1, 0) + α(1, 0) = (3, 1) + (3, 1) = (1, 2). 4
5 13. ( i, (1 2 3) ) 14. ( (1 2)(3 4), 3 ) 15. D 4 has 5 elements of order 2, and S 3 has 2 elements of order 3, so D 4 S 3 has 10 elements of order Each Z 9 has 6 elements of order 9, and 3 elements of order 1 or 3. Therefore Z 9 Z 9 has = 72 elements of order (a) Z 6 Z 10 (b) Z 2 Z 4 Z 10 (c) Z 4 Z 18 (d) Z 6 Z 2 Z Aut(Z 45 ) U(45) U(5) U(9) Z 4 Z Aut ( U(25) ) Aut(Z 20 ) U(20) U(4) U(5) Z 2 Z Since U(900) U(4) U(9) U(25) Z 2 Z 6 Z 20, the largest possible order is lcm(2, 6, 20) = Z 12, Z 2 Z 6, D 6, A Z 30, D 15, D 3 Z 5, D 5 Z 3 (Note that the last three groups all have a different number of elements of order 2, and are therefore not isomorphic.) 23. (a) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, {(1 2 3), (1 3 4), (2 4 3), (1 4 2)}, {(1 2 4), (1 4 3), (1 3 2), (2 3 4)} (b) {1, 11}, {7 17}, {13, 23}, {19, 29} 25. {(0, 0, 0), (1, 1, 1)}, {(1, 0, 0), (0, 1, 1)}, {(0, 1, 0), (1, 0, 1)}, {(0, 0, 1), (1, 1, 0)} 26. Let H be a proper subgroup of G. By Lagrange s theorem H must be a factor of pq, so H {1, p, q, pq}. Since H is a proper subgroup, we know that H pq. Therefore H either has prime order or is trivial, which in either case implies that H is cyclic. 27. (a) 6 (b) 2 (c) lcm(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) = = (a) {e, (2 4)(5 6)} (b) {1, 2, 3, 4} (c) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} (d) {5, 6} /4 = 15. 5
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