HOMEWORK Graduate Abstract Algebra I May 2, 2004


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1 Math 5331 Sec 121 Spring 2004, UT Arlington HOMEWORK Graduate Abstract Algebra I May 2, 2004 The required text is Algebra, by Thomas W. Hungerford, Graduate Texts in Mathematics, Vol 73, Springer. (it will be denoted by [H] in this course); but other books that might be helpful are Algebra Vol 1, 2,..., by P. M. Cohn, J. Wiley and Sons, Ltd. Introduction to Group Theory, by W. Ledermann, Longman Group Ltd. Lectures in Abstract Algebra, by N. Jacobson, Graduate Texts in Mathematics, Vol 30, 31, 32, Springer Algebra, by S. Lang, AddisonWesley. Algebra, by M. Artin, Prentice Hall, In the following questions, C denotes the complex numbers, R denotes the real numbers, Z denotes the set of all integers, and N denotes the set of positive integers. 1. Read pages in [H]. 2. Prove that GL(n, R) is not closed under matrix addition. 3. Prove that Z is a group under +. (This group is an example of an infinite cyclic group.) 4. Prove that {0, 1, 2, 3,..., n 1} is a group under + modulo n. (This group is an example of a finite cyclic group.) 5. Prove that {e, a, a 2, a 3,..., a n 1 }, where a n = e, is a group under multiplication. (This group is an example of a finite cyclic group.) [ ] [ ] i Let A = and B =, where i 0 i = 1. Find A 4 and B 2. Prove that {I, A, A 2, A 3, B, AB, A 2 B, A 3 B} is a group. (This group is called the quaternion group.) 7. Let S = {e, a, b, c, d} with multiplication given by the multiplication table: e a b c d e e a b c d a a e d b c b b c e d a c c d a e b d d b c a e. Verify that S is NOT a group. (Hint: consider abc.) 8. [H, Page 29, #5] Prove that S n = n!. 1
2 9. Let G and H be groups. (a) Prove that G H is a group. (b) Prove that if G and H are each abelian, then so is G H. 10. [H, Page 29, #6] Let Z 2 denote the group {0, 1} under + modulo 2. Write out the elements of Z 2 Z 2 and an addition table for the elements. (Z 2 Z 2 is called the Klein four group.) 11. Verify that {1, 2,..., 9} is NOT a group under multiplication modulo 10 (i.e., multiply, then reduce product modulo 10). 12. Prove that a group G is abelian if and only if (ab) 2 = a 2 b 2 for all a, b G. 13. [H, Page 30, #13] Let G be a group. Prove that if a 2 = e for all a G, then G is abelian. 14. Prove that a finite group of even order contains an odd number of elements of order two (i.e., elements a e of the form a 2 = e). (Hint: any element and its inverse have the same order.) 15. The quaternion numbers are elements of the form a + bi + cj + dk where a, b, c, d R and i 2 = j 2 = 1, ij = ji = k. Let G denote the set consisting of i, j and all possible products of i and j. Prove that G is a group. How similar is this group to the group presented in Question 6? Be explicit in your comparison. 16. Read and understand [H, page 5, Theorem 3.1] and the proof thereof. 17. Read pages in [H]. 18. [H, Page 33, #2] Prove that a group G is abelian if and only if the map G G given by x x 1 is an automorphism. 19. Do [H, Page 33, #3] and prove that the group therein is isomorphic to the group in Question Prove that the group in Question 6 is isomorphic to the group in Question [H, Page 34, #7] Show that if m Z is a fixed nonzero integer, then the set {nm : n Z} is an additive subgroup of Z and that it is is isomorphic to Z. 22. Find all subgroups of the dihedral group D 4. (Note: find means more than list justify your answer by showing your work.) 23. [H, Page 34, #10] Find all subgroups of Z 2 Z 2. Is Z 2 Z 2 isomorphic to Z 4? (Remember to justify your work!) 24. [H, Page 34, #9] Let φ : G H be a homomorphism of groups, G < G and H < H. (a) Prove that φ(g ) is a subgroup of H. (b) Prove that the preimage of H is a subgroup of G. (Note: Ker φ was proved in class to be a subgroup of G.) 2
3 25. [H, Page 34, #11] Prove that if G is a group, then the set Z = {a G : ag = ga for all g G} is an abelian subgroup of G. (Z is called the center of G.) 26. [H, Page 34, #13] Suppose G = g is a cyclic group and that H is any group. Prove that every homomorphism φ : G H is completely determined by the element φ(g) H. 27. [H, Page 33, #5] Let S denote a nonempty subset of a group G and define a relation on G by a b if and only if ab 1 S. Show that is an equivalence relation if and only if S is a subgroup of G. 28. [H, Page 34, #17] Let G be an abelian group and let H and K be subgroups of G. Show that the join H K is the set {hk : h H, k K}. 29. [H, Page 36, #1] Let a and b be elements of any group G. Show that a = a 1, ab = ba and that c 1 ac = a for all c G. 30. Let G be an abelian group containing elements a and b of orders m and n respectively, where gcd(m, n) = 1. Show that G contains an element whose order is mn. 31. [H, Page 36, #3] Let G be an abelian group of order pq, with gcd(p, q) = 1. Show that if there exist elements a, b G such that a = p and b = q, then G is cyclic. 32. [H, Page 36, #4] If φ : G H is a group homomorphism, g G and φ(g) <, then either g = or φ(g) divides g. 33. [H, Page 37, #5] Let G be [ the multiplicative ] group of all nonsingular [ 2 2 matrices ] with rational entries. Show that A = has order 4 and that B = has order 3, but that AB has infinite order. Conversely, show that the additive group Z 2 Z contains nonzero elements a, b of infinite order such that a + b has finite order. 34. [H, Page 37, #6] Let G be a cyclic group of order n. Prove that if k n, then G has exactly one subgroup of order k. 35. Read pages in [H]. 36. [H, Page 40, #2] (a) Let H be the cyclic subgroup (of order 2) of S 3 generated by. Show that no left coset of H (except H itself) is also a right coset, and that there exists a S 3 such that ah Ha = {a}. (b) If K is the cyclic subgroup (of order 3) of S 3 generated by left coset of K is also a right coset of K , show that every
4 37. [H, Page 40, #3] Show that the following conditions on a finite group G are equivalent. (a) G is prime (b) G e and G has no proper subgroups (c) G = Z p for some prime p. 38. [H, Page 40, #5] Prove that there are only two distinct groups of order 4 (up to isomorphism), namely Z 4 and Z 2 Z 2. (Hint: by Lagrange s Theorem, a group of order 4 that is not cyclic must consist of an identity and three elements of order 2.) Let G = S 3, H = σ where σ = and K = τ where τ = that HK is not a subgroup of G Show [H, Page 40, #11] Let G be a group of order 2n. Show that G contains an element of order 2. Also show that if n is odd and G abelian, then there is only one element of order 2. (Hint: consider Question 14 above.) 41. [H, Page 41, #14] Let G be a group and a, b G such that (a) a = 4 = b (b) a 2 = b 2 (c) ba = a 3 b = a 1 b (d) a b (e) G = a, b. Show that G = 8 and that G is isomorphic to the quaternion group from Questions 6 and Read pages in [H]. 43. [H, Page 46, #17] (a) Consider the subgroups 6 and 30 of Z and show that 6 / 30 = Z 5. (b) For any k, m > 0, show that k / km = Z m, and that Z/ m = 1 / m = Z m. 44. [H, Page 45, #5] Let N < S 4 consist of all those permutations σ such that σ(4) = 4. Is N normal in S 4? 45. [H, Page 45, #1] If N is a subgroup of index 2 in a group G, prove that N is normal in G. 46. [H, Page 45, #2] If {N i : i I} is a family of normal subgroups of a group G, then prove that i I N i is a normal subgroup of G. 47. [H, Page 45, #6] Prove that if H < G, then the set aha 1 is a subgroup for each a G, and that H = aha [H, Page 45, #7] Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, prove that H is normal in G. (Hint: consider Question 47.) 49. [H, Page 45, #9(a)] Prove that the center of any group G is a normal subgroup of G. 4
5 50. [H, Page 45, #10] Find subgroups H and K of the dihedral group D 4 such that H K and K D 4, but H is not normal in D [H, Page 45, #11] If H is a cyclic subgroup of a group G such that H G, prove that every subgroup of H is normal in G. 52. [H, Page 46, #12] If G is a group and H G and both H and G/H are finitely generated, prove that G is also finitely generated. 53. [H, Page 46, #16] Let G and H be groups and assume that H is abelian. If φ : G H is a homomorphism, such that Ker φ is contained in a subgroup N < G, prove that N G. 54. Read pages in [H] Factor σ = into a product of mutually disjoint cycles. Find the orbits of σ and find σ. Compute sgn(σ) Factor τ = into a product of mutually disjoint cycles. Find the orbits of τ and find τ. Compute sgn(τ). a b c d e f 57. Factor µ = into a product of mutually disjoint cycles. Find the orbits of µ c e d f b a and find µ. Compute sgn(µ). 58. Find the orbits of ( )(1 6) and the orbits of ( )( ). 59. Verify that {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} is an abelian group of order four. To which group of order four is it isomorphic? 60. [H, Page 51, #5] Let σ, τ S n. Prove that if σ is even (respectively, odd), then so is τστ We say that σ, τ S n are disjoint if σ(x) x τ(x) = x, and if τ(y) y σ(y) = y (note: there might exist z such that σ(z) = z = τ(z)). Prove that if σ and τ are disjoint and στ = e, then σ = e = τ. 62. Let H and K be groups. Show that (a) (h, e) H {e} H K and (e, k) {e} K H K commute; (b) H {e} and {e} K are normal subgoups of H K; (c) (H {e}) ({e} K) = {e} and (H {e})({e} K) = H K. 63. Show that Z 6 = Z2 Z Prove that if p is a prime, then Z p 2 = Zp Z p. 65. Prove that if H and K are groups, then H K H {e} = K. 5
6 66. Show that it is possible for a group G to contain three distinct normal subgroups H, K and L such that G = H L = K L; that is, HL = G = KL and H L = {e} = K L. (Hint: try the Kleinfour group.) 67. Prove that an abelian group G of order p 2, where p is prime, is either cyclic or isomorphic to Z p Z p. 68. Read pages in [H]. 69. [H, Page 68, #1] Let g a free group. Prove that if g e, then g has infinite order. (Hint: Use Theorem 9.2 in [H], which is the definition given in class of a free group.) 70. Show that a, b a 1 ba = b 2, b 1 ab = a 2 = {e}. 71. [H, Page 69, #5] Let G = a, b a 8 = b 2 a 4 = ab 1 ab = e. Show that G [H, Page 69, #6] Let G = a, b a 2 = b 3 = a 1 b 1 ab = e. Show that G = Z [H, Page 69, #7] Let G = a, b a 2 = b 3 = e. Show that G is infinite and nonabelian. 74. Read pages in [H]. 75. (a) Show that an abelian free group must be a cyclic group. (b) Give an example of a free abelian group that is not a cyclic group. 76. Consider H = 4, 6 = 4Z + 6Z < Z. Show that the set {4, 6} is not a basis of H and that it does not contain a basis of H. Show that H is a free abelian group and find rank(h). 77. [H, Page 74, #1] (a) If G is an abelian group and m Z, prove that mg = {mg : g G} is a subgroup of G. (b) If G i are abelian groups, where i I, and if G = i I G i, prove that mg = i I mg i and that G/mG = i I G i/mg i. 78. [H, Page 74, #2(a)(c)] A subset X of an abelian group G is said to be linearly independent if n 1 x 1 + +n k x k = 0 always implies that n i = 0 for all i (where n i Z and x 1,..., x k are distinct elements of X). (a) Prove that X is linearly independent if and only if every nonzero element of the subgroup X may be written uniquely in the form n 1 x 1 + +n k x k (where n i Z, n i 0, x 1,..., x k distinct elements of X). (b) If G is a free abelian group of rank n, show that it is not true that every linearly independent subset of n elements is a basis. (Hint: consider G = Z and lecture notes.) (c) If G is a free abelian group, show that it is not true that every linearly independent subset of G may be extended to a basis of G. 6
7 79. In this question, we will give an alternative justification for why the rank of a finitely generated free abelian group is well defined. Let G be a finitely generated abelian group and suppose that G has bases {x 1,..., x n } and {y 1,..., y m }. We wish to prove that m = n. (a) Prove that, for each i = 1,..., n, x i = m j=1 α ijy j, for some α ij Z, and that, for each j = 1,..., m, y j = n k=1 β jkx k, for some β jk Z. (b) Prove that x i = m n j=1 k=1 α ijβ jk x k, for all i = 1,..., n. (c) Prove that { m 0 if k i j=1 α ijβ jk = δ ik = i.e., that the matrix product (α 1 if k = i, ij )(β jk ) = I n ( ), the n n identity matrix. (d) Prove that y j = n m k=1 r=1 β jkα kr y r, for all j = 1,..., m, and that n { k=1 β jkα kr = δ jr = 0 if r j i.e., that the matrix product (β 1 if r = j, jk )(α kr ) = I m ( ), the m m identity matrix. (e) By computing the sum of the diagonal elements in ( ) and in ( ) respectively, prove that m = n. 80. Suppose that G is a free abelian group and that {x 1,..., x n } is a basis of G, where n 2. Prove that the subgroup H = x 1,..., x n 1 is a free abelian group of rank = n Suppose that G is a free abelian group. (a) If G has basis {x, y}, prove that G = x y. (b) If G has basis {x 1,..., x n }, prove that G = x 1 x 2,..., x n. 82. If G is a free abelian group, finitely generated by n elements, prove that rank(g) n. (From Question 76, recall that a generating set need not contain a basis; instead look at Page 74 of [H].) 83. [H, Page 75, #7] Use [H, Page 75, #6] to prove that a nonzero abelian group has a subgroup of index n for all n N. 84. Read/skim pages in [H]. 85. Prove [H, Page 77, Lemma II2.5(i)(v)]. 86. [H, Page 82, #12] (a) What are the elementary divisors of the group Z 2 Z 9 Z 35 ; what are its invariant factors? Do the same for Z 26 Z 42 Z 49 Z 200 Z (b) Determine, up to isomorphism, all abelian groups of order 64; do the same for order 96. (c) Determine, up to isomorphism, all abelian groups of order n for n [H, Page 82, #13] Show that the invariant factors of Z m Z n are gcd(m, n) and [m, n] (the latter being the least common multiple) if gcd(m, n) > 1, but only mn if gcd(m, n) = 1. (Hint: recall Questions 30 & 31.) 88. Let G = Z 2 Z 4 Z 27. For all H G, prove that G contains a subgroup that is isomorphic to G/H. 7
8 89. [H, Page 81, #1] Show that a finite abelian group that is not cyclic contains a subgroup which is isomorphic to Z p Z p for some prime p. 90. [H, Page 82, #6] Let k, m N \ {0}. If gcd(k, m) = 1, show that kz m = Z m and Z m [k] = 0. If k m, say m = kd, show that kz m = Zd and Z m [k] = Z k. 91. (a) How many subgroups of order 4 does Z 4 Z 8 have? (b) How many subgroups of order 25 does Z 25 Z 125 have? (c) [H, Page 82, #9] How many subgroups of order p 2 does Z p 2 Z p 3 have, where p is a prime? 92. Show that S 2 and S 3 are indecomposable. 93. Read pages in [H]. 94. Prove Theorem 4.2 on page 89 in [H]. 95. Let G be a group and let S denote the set of subgroups of G. Suppose K S. (a) Prove K N G (K). (b) Prove K G if and only if N G (K) = G. 96. [H, Page 92, #4] Let H be a subgroup of G. The centralizer of H is the set C G (H) = {g G : hg = gh for all h H}. Prove that C G (H) N G (H). 97. [H, Page 92, #6(a),(b),(d)] Let G be a group acting on a set S that contains at least two elements. Assume that G is transitive; that is, given any x and y S, there exists g G such that gx = y. Prove that (a) for x S, the orbit x of x is S; (b) the stabilizers G x, where x S, are all conjugate; (c) for x S, S = [G : G x ], hence S divides G. 98. If G acts on itself by left translation, show the action is transitive, and if g G, then ḡ = G. 99. [H, Page 92, #8] Exhibit an automorphism of Z 6 that is not an inner automorphism. (Hint: recall Question 26.) 100. [H, Page 92, #14] Suppose that G is a group and that G = pn, where p is prime, p > n. Prove that if H G, with H = p, then H G. (Hint: consider [H, Prop. II.4.8].) 101. Suppose G is a group such that G = p n, where p is prime and n N \ {0}. Prove that the center Z e. (Hint: use the class equation.) 102. Suppose G is a group such that G = p 2, where p is prime. Prove that G is abelian. (Hint: consider Question 101 and the last question on Midterm 1.) 103. Read pages in [H] Prove Corollary 5.8 on page 95 in [H]. 8
9 105. [H, Page 96, #1] If N G and if N and G/N are both pgroups, prove that G is a pgroup too [H, Page 96, #5] If P is a normal Sylow psubgroup of a finite group G and if f : G G is an endomorphism, prove that f(p ) P [H, Page 96, #6] If H is a normal subgroup of order p k of a finite group G, prove that H is contained in every Sylow psubgroup of G [H, Page 96, #7] Find the Sylow 2subgroups and the Sylow 3subgroups of S 3 and S [H, Page 96, #9] If G = p n q, with p and q primes and p > q, prove that G contains a unique normal subgroup of index q [H, Page 96, #10] Show that every group of order 12, 28, 56 and 200 contains a normal Sylow subgroup, and so is not simple Prove that there does NOT exist a simple group of order Read pages in [H] Let G be a group of order p 2 q, where p and q are prime, q < p and q does not divide p 2 1. Prove that G is abelian. (Hint: show that there exists a unique Sylow psubgroup P, and a unique Sylow qsubgroup Q and show that G = P Q.) 114. Read page 100 in [H] and skim pages in [H] [H, Page 106, #1] (a) Prove that A 4 is not the direct product of its Sylow subgroups, but that A 4 has the property that mn = 12 = A 4 and (m, n) = 1, implies that A 4 has a subgroup of order m. (b) Show that S 3 has subgroups of orders 1, 2, 3 and 6, but prove that S 3 is not the direct product of its Sylow subgroups [H, Page 106, #2] Let G be a group and a, b G. Denote the commutator aba 1 b 1 G by [a, b]. Show that for any a, b, c G, [ab, c] = a[b, c]a 1 [a, c] [H, Page 107, #9] Show that the commutator subgroup of S 4 is A 4. What is the commutator subgroup of A 4? 118. [H, Page 107, #10] Show that S n is solvable for all n 4, but that S 3 and S 4 are not nilpotent [H, Page 107, #14] Prove that if N G and N G = {e}, then N < Z(G) Find all the composition factors of S 4, D 6 and the quaternion group (given in Questions 6 & 15) respectively. END HOMEWORK 9
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