HOMEWORK Graduate Abstract Algebra I May 2, 2004


 Karen Park
 1 years ago
 Views:
Transcription
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
PROBLEMS FROM GROUP THEORY
PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationAlgebra Exercises in group theory
Algebra 3 2010 Exercises in group theory February 2010 Exercise 1*: Discuss the Exercises in the sections 1.11.3 in Chapter I of the notes. Exercise 2: Show that an infinite group G has to contain a nontrivial
More informationA. (Groups of order 8.) (a) Which of the five groups G (as specified in the question) have the following property: G has a normal subgroup N such that
MATH 402A  Solutions for the suggested problems. A. (Groups of order 8. (a Which of the five groups G (as specified in the question have the following property: G has a normal subgroup N such that N =
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationAlgebra. Travis Dirle. December 4, 2016
Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................
More informationSchool of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and ReActivation
MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and ReActivation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK
More informationALGEBRA QUALIFYING EXAM PROBLEMS
ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General
More informationAssigment 1. 1 a b. 0 1 c A B = (A B) (B A). 3. In each case, determine whether G is a group with the given operation.
1. Show that the set G = multiplication. Assigment 1 1 a b 0 1 c a, b, c R 0 0 1 is a group under matrix 2. Let U be a set and G = {A A U}. Show that G ia an abelian group under the operation defined by
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of
More informationA Little Beyond: Linear Algebra
A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of nonzero polynomials in [x], no two
More informationMA441: Algebraic Structures I. Lecture 26
MA441: Algebraic Structures I Lecture 26 10 December 2003 1 (page 179) Example 13: A 4 has no subgroup of order 6. BWOC, suppose H < A 4 has order 6. Then H A 4, since it has index 2. Thus A 4 /H has order
More informationExtra exercises for algebra
Extra exercises for algebra These are extra exercises for the course algebra. They are meant for those students who tend to have already solved all the exercises at the beginning of the exercise session
More informationABSTRACT ALGEBRA: REVIEW PROBLEMS ON GROUPS AND GALOIS THEORY
ABSTRACT ALGEBRA: REVIEW PROBLEMS ON GROUPS AND GALOIS THEORY John A. Beachy Northern Illinois University 2000 ii J.A.Beachy This is a supplement to Abstract Algebra, Second Edition by John A. Beachy and
More informationAlgebra Ph.D. Entrance Exam Fall 2009 September 3, 2009
Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Directions: Solve 10 of the following problems. Mark which of the problems are to be graded. Without clear indication which problems are to be graded
More information120A LECTURE OUTLINES
120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication
More informationAbstract Algebra: Supplementary Lecture Notes
Abstract Algebra: Supplementary Lecture Notes JOHN A. BEACHY Northern Illinois University 1995 Revised, 1999, 2006 ii To accompany Abstract Algebra, Third Edition by John A. Beachy and William D. Blair
More informationMath 210A: Algebra, Homework 5
Math 210A: Algebra, Homework 5 Ian Coley November 5, 2013 Problem 1. Prove that two elements σ and τ in S n are conjugate if and only if type σ = type τ. Suppose first that σ and τ are cycles. Suppose
More informationMath 546, Exam 2 Information.
Math 546, Exam 2 Information. 10/21/09, LC 303B, 10:1011:00. Exam 2 will be based on: Sections 3.2, 3.3, 3.4, 3.5; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/546fa09/546.html)
More informationAbstract Algebra II Groups ( )
Abstract Algebra II Groups ( ) Melchior Grützmann / melchiorgfreehostingcom/algebra October 15, 2012 Outline Group homomorphisms Free groups, free products, and presentations Free products ( ) Definition
More informationAlgebra SEP Solutions
Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since
More informationKevin James. pgroups, Nilpotent groups and Solvable groups
pgroups, Nilpotent groups and Solvable groups Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G. Definition A maximal subgroup of a
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationCourse 311: Abstract Algebra Academic year
Course 311: Abstract Algebra Academic year 200708 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 1 Topics in Group Theory 1 1.1 Groups............................... 1 1.2 Examples of Groups.......................
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationSUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More information1. Group Theory Permutations.
1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7
More informationSUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III Week 1 Lecture 1 Tuesday 3 March.
SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III 2009 Week 1 Lecture 1 Tuesday 3 March. 1. Introduction (Background from Algebra II) 1.1. Groups and Subgroups. Definition 1.1. A binary operation on a set
More informationInternational Journal of Pure and Applied Mathematics Volume 13 No , MGROUP AND SEMIDIRECT PRODUCT
International Journal of Pure and Applied Mathematics Volume 13 No. 3 2004, 381389 MGROUP AND SEMIDIRECT PRODUCT Liguo He Department of Mathematics Shenyang University of Technology Shenyang, 110023,
More informationCSIR  Algebra Problems
CSIR  Algebra Problems N. Annamalai DST  INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli 620024 Email: algebra.annamalai@gmail.com Website: https://annamalaimaths.wordpress.com
More informationElements of solution for Homework 5
Elements of solution for Homework 5 General remarks How to use the First Isomorphism Theorem A standard way to prove statements of the form G/H is isomorphic to Γ is to construct a homomorphism ϕ : G Γ
More informationGroups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems
Group Theory Groups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems Groups Definition : A nonempty set ( G,*)
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
More informationPh.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018
Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018 Do 6 problems with at least 2 in each section. Group theory problems: (1) Suppose G is a group. The
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationMath 594, HW2  Solutions
Math 594, HW2  Solutions Gilad Pagi, Feng Zhu February 8, 2015 1 a). It suffices to check that NA is closed under the group operation, and contains identities and inverses: NA is closed under the group
More informationCHARACTER THEORY OF FINITE GROUPS. Chapter 1: REPRESENTATIONS
CHARACTER THEORY OF FINITE GROUPS Chapter 1: REPRESENTATIONS G is a finite group and K is a field. A Krepresentation of G is a homomorphism X : G! GL(n, K), where GL(n, K) is the group of invertible n
More informationSolutions to Assignment 4
1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2
More informationFall /29/18 Time Limit: 75 Minutes
Math 411: Abstract Algebra Fall 2018 Midterm 10/29/18 Time Limit: 75 Minutes Name (Print): Solutions JHUID: This exam contains 8 pages (including this cover page) and 6 problems. Check to see if any pages
More informationits image and kernel. A subgroup of a group G is a nonempty subset K of G such that k 1 k 1
10 Chapter 1 Groups 1.1 Isomorphism theorems Throughout the chapter, we ll be studying the category of groups. Let G, H be groups. Recall that a homomorphism f : G H means a function such that f(g 1 g
More informationSection III.15. FactorGroup Computations and Simple Groups
III.15 FactorGroup Computations 1 Section III.15. FactorGroup Computations and Simple Groups Note. In this section, we try to extract information about a group G by considering properties of the factor
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.
MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is
More informationSF2729 GROUPS AND RINGS LECTURE NOTES
SF2729 GROUPS AND RINGS LECTURE NOTES 20110301 MATS BOIJ 6. THE SIXTH LECTURE  GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking
More informationMath 451, 01, Exam #2 Answer Key
Math 451, 01, Exam #2 Answer Key 1. (25 points): If the statement is always true, circle True and prove it. If the statement is never true, circle False and prove that it can never be true. If the statement
More informationPermutation groups H. W. Lenstra, Fall streng/permutation/index.html
Permutation groups H. W. Lenstra, Fall 2007 http://www.math.leidenuniv.nl/ streng/permutation/index.html Solvable groups. Let G be a group. We define the sequence G (0) G (1) G (2)... of subgroups of G
More information3. G. Groups, as men, will be known by their actions.  Guillermo Moreno
3.1. The denition. 3. G Groups, as men, will be known by their actions.  Guillermo Moreno D 3.1. An action of a group G on a set X is a function from : G X! X such that the following hold for all g, h
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationGroup Theory
Group Theory 2014 2015 Solutions to the exam of 4 November 2014 13 November 2014 Question 1 (a) For every number n in the set {1, 2,..., 2013} there is exactly one transposition (n n + 1) in σ, so σ is
More informationB Sc MATHEMATICS ABSTRACT ALGEBRA
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z
More informationDISCRETE MATH (A LITTLE) & BASIC GROUP THEORY  PART 3/3. Contents
DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY  PART 3/3 T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Cayley s Theorem 1 2. The permutation group S n 2 3. Center of a group, and centralizers 4 4. Group actions
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationBasic Definitions: Group, subgroup, order of a group, order of an element, Abelian, center, centralizer, identity, inverse, closed.
Math 546 Review Exam 2 NOTE: An (*) at the end of a line indicates that you will not be asked for the proof of that specific item on the exam But you should still understand the idea and be able to apply
More informationCosets, factor groups, direct products, homomorphisms, isomorphisms
Cosets, factor groups, direct products, homomorphisms, isomorphisms Sergei Silvestrov Spring term 2011, Lecture 11 Contents of the lecture Cosets and the theorem of Lagrange. Direct products and finitely
More informationTheorems and Definitions in Group Theory
Theorems and Definitions in Group Theory Shunan Zhao Contents 1 Basics of a group 3 1.1 Basic Properties of Groups.......................... 3 1.2 Properties of Inverses............................. 3
More informationName: Solutions  AI FINAL EXAM
1 2 3 4 5 6 7 8 9 10 11 12 13 total Name: Solutions  AI FINAL EXAM The first 7 problems will each count 10 points. The best 3 of # 813 will count 10 points each. Total is 100 points. A 4th problem from
More informationMATH HL OPTION  REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis
MATH HL OPTION  REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.
More informationAlgebra Exam, Spring 2017
Algebra Exam, Spring 2017 There are 5 problems, some with several parts. Easier parts count for less than harder ones, but each part counts. Each part may be assumed in later parts and problems. Unjustified
More informationIntroduction to Groups
Introduction to Groups HongJian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)
More informationAlgebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001
Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationAbstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications
1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the
More informationOhio State University Department of Mathematics Algebra Qualifier Exam Solutions. Timothy All Michael Belfanti
Ohio State University Department of Mathematics Algebra Qualifier Exam Solutions Timothy All Michael Belfanti July 22, 2013 Contents Spring 2012 1 1. Let G be a finite group and H a nonnormal subgroup
More informationCCharacteristically Simple Groups
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 35(1) (2012), 147 154 CCharacteristically Simple Groups M. Shabani Attar Department
More informationHomework Problems, Math 200, Fall 2011 (Robert Boltje)
Homework Problems, Math 200, Fall 2011 (Robert Boltje) Due Friday, September 30: ( ) 0 a 1. Let S be the set of all matrices with entries a, b Z. Show 0 b that S is a semigroup under matrix multiplication
More informationAlgebraic structures I
MTH5100 Assignment 110 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More informationHigher Algebra Lecture Notes
Higher Algebra Lecture Notes October 2010 Gerald Höhn Department of Mathematics Kansas State University 138 Cardwell Hall Manhattan, KS 665062602 USA gerald@math.ksu.edu This are the notes for my lecture
More informationINTRODUCTION TO THE GROUP THEORY
Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc email: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher
More informationMATH 420 FINAL EXAM J. Beachy, 5/7/97
MATH 420 FINAL EXAM J. Beachy, 5/7/97 1. (a) For positive integers a and b, define gcd(a, b). (b) Compute gcd(1776, 1492). (c) Show that if a, b, c are positive integers, then gcd(a, bc) = 1 if and only
More informationExercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups
Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum
More informationANALYSIS OF SMALL GROUPS
ANALYSIS OF SMALL GROUPS 1. Big Enough Subgroups are Normal Proposition 1.1. Let G be a finite group, and let q be the smallest prime divisor of G. Let N G be a subgroup of index q. Then N is a normal
More informationAlgebra Qualifying Exam, Fall 2018
Algebra Qualifying Exam, Fall 2018 Name: Student ID: Instructions: Show all work clearly and in order. Use full sentences in your proofs and solutions. All answers count. In this exam, you may use the
More informationSchool of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information
MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon
More informationFrank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups:
Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups: Definition: The external direct product is defined to be the following: Let H 1,..., H n be groups. H 1 H 2 H n := {(h 1,...,
More informationRepresentation Theory
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character
More informationTwo subgroups and semidirect products
Two subgroups and semidirect products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset
More informationYale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall Midterm Exam Review Solutions
Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall 2015 Midterm Exam Review Solutions Practice exam questions: 1. Let V 1 R 2 be the subset of all vectors whose slope
More informationChapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups  groups of permutations
Chapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups  groups of permutations (bijections). Definition A bijection from a set A to itself is also
More informationLecture 20 FUNDAMENTAL Theorem of Finitely Generated Abelian Groups (FTFGAG)
Lecture 20 FUNDAMENTAL Theorem of Finitely Generated Abelian Groups (FTFGAG) Warm up: 1. Let n 1500. Find all sequences n 1 n 2... n s 2 satisfying n i 1 and n 1 n s n (where s can vary from sequence to
More informationLecture 5.6: The Sylow theorems
Lecture 5.6: The Sylow theorems Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 5.6:
More informationENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.
ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.
More informationMAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems.
MAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems. Problem 1 Find all homomorphisms a) Z 6 Z 6 ; b) Z 6 Z 18 ; c) Z 18 Z 6 ; d) Z 12 Z 15 ; e) Z 6 Z 25 Proof. a)ψ(1)
More informationAlgebraI, Fall Solutions to Midterm #1
AlgebraI, Fall 2018. Solutions to Midterm #1 1. Let G be a group, H, K subgroups of G and a, b G. (a) (6 pts) Suppose that ah = bk. Prove that H = K. Solution: (a) Multiplying both sides by b 1 on the
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More informationThe Outer Automorphism of S 6
Meena Jagadeesan 1 Karthik Karnik 2 Mentor: Akhil Mathew 1 Phillips Exeter Academy 2 Massachusetts Academy of Math and Science PRIMES Conference, May 2016 What is a Group? A group G is a set of elements
More informationCONSEQUENCES OF THE SYLOW THEOREMS
CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.
More informationSolutions of exercise sheet 4
DMATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 4 The content of the marked exercises (*) should be known for the exam. 1. Prove the following two properties of groups: 1. Every
More informationMATH 28A MIDTERM 2 INSTRUCTOR: HAROLD SULTAN
NAME: MATH 28A MIDTERM 2 INSTRUCTOR: HAROLD SULTAN 1. INSTRUCTIONS (1) Timing: You have 80 minutes for this midterm. (2) Partial Credit will be awarded. Please show your work and provide full solutions,
More informationREU 2007 Discrete Math Lecture 2
REU 2007 Discrete Math Lecture 2 Instructor: László Babai Scribe: Shawn Drenning June 19, 2007. Proofread by instructor. Last updated June 20, 1 a.m. Exercise 2.0.1. Let G be an abelian group and A G be
More informationCS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a
Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 20178: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationMath 2070BC Term 2 Weeks 1 13 Lecture Notes
Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic
More informationREPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012
REPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012 JOSEPHINE YU This note will cover introductory material on representation theory, mostly of finite groups. The main references are the books of Serre
More informationMath 370 Spring 2016 Sample Midterm with Solutions
Math 370 Spring 2016 Sample Midterm with Solutions Contents 1 Problems 2 2 Solutions 5 1 1 Problems (1) Let A be a 3 3 matrix whose entries are real numbers such that A 2 = 0. Show that I 3 + A is invertible.
More informationNotes on Group Theory. by Avinash Sathaye, Professor of Mathematics November 5, 2013
Notes on Group Theory by Avinash Sathaye, Professor of Mathematics November 5, 2013 Contents 1 Preparation. 2 2 Group axioms and definitions. 2 Shortcuts................................. 2 2.1 Cyclic groups............................
More informationMath 581 Problem Set 8 Solutions
Math 581 Problem Set 8 Solutions 1. Prove that a group G is abelian if and only if the function ϕ : G G given by ϕ(g) g 1 is a homomorphism of groups. In this case show that ϕ is an isomorphism. Proof:
More informationMath 3140 Fall 2012 Assignment #3
Math 3140 Fall 2012 Assignment #3 Due Fri., Sept. 21. Remember to cite your sources, including the people you talk to. My solutions will repeatedly use the following proposition from class: Proposition
More information17 More Groups, Lagrange s Theorem and Direct Products
7 More Groups, Lagrange s Theorem and Direct Products We consider several ways to produce groups. 7. The Dihedral Group The dihedral group D n is a nonabelian group. This is the set of symmetries of a
More informationGroups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group
C H A P T E R t h r e E Groups Introduction Some of the standard topics in elementary group theory are treated in this chapter: subgroups, cyclic groups, isomorphisms, and homomorphisms. In the development
More information(1) Let G be a finite group and let P be a normal psubgroup of G. Show that P is contained in every Sylow psubgroup of G.
(1) Let G be a finite group and let P be a normal psubgroup of G. Show that P is contained in every Sylow psubgroup of G. (2) Determine all groups of order 21 up to isomorphism. (3) Let P be s Sylow
More information