Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 5, Problem 60E
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Chapter 5 Solutions
Contemporary Abstract Algebra
Ch. 5 - Let [123456213546]and=[123456612435] . Compute...Ch. 5 - Let [1234567823451786]and=[1234567813876524] ....Ch. 5 - Write each of the following permutations as a...Ch. 5 - Find the order of each of the following...Ch. 5 - What is the order of each of the following...Ch. 5 - What is the order of each of the following...Ch. 5 - What is the order of the product of a pair of...Ch. 5 - Determine whether the following permutations are...Ch. 5 - What are the possible orders for the elements of...Ch. 5 - Show that A8 contains an element of order 15.
Ch. 5 - Find an element in A12 of order 30.Ch. 5 - Show that a function from a finite set S to itself...Ch. 5 - Prob. 13ECh. 5 - Suppose that is a 6-cycle and is a 5-cycle....Ch. 5 - Prob. 15ECh. 5 - If is even, prove that 1 is even. If is odd,...Ch. 5 - Prob. 17ECh. 5 - In Sn , let be an r-cycle, an s-cycle, and a...Ch. 5 - Let and belong to Sn . Prove that is even if...Ch. 5 - Associate an even permutation with the number +1...Ch. 5 - Complete the following statement: A product of...Ch. 5 - What cycle is (a1a2an)1 ?Ch. 5 - Show that if H is a subgroup of Sn , then either...Ch. 5 - Suppose that H is a subgroup of Sn of odd order....Ch. 5 - Give two reasons why the set of odd permutations...Ch. 5 - Let and belong to Sn . Prove that 11 is an...Ch. 5 - Prob. 27ECh. 5 - How many elements of order 5 are in S7 ?Ch. 5 - Prob. 29ECh. 5 - Prove that (1234) is not the product of 3-cycles....Ch. 5 - Let S7 and suppose 4=(2143567) . Find . What are...Ch. 5 - My mind rebels at stagnation. Give me problems,...Ch. 5 - Let (a1a2a3a4)and(a5a6) be disjoint cycles in S10...Ch. 5 - If and are distinct 2-cycles, what are the...Ch. 5 - Prob. 35ECh. 5 - Let =(1,3,5,7,9,8,6)(2,4,10) . What is the...Ch. 5 - Prob. 37ECh. 5 - Let H=S5(1)=1and(3)=3 . Prove that H is a...Ch. 5 - In S4 , find a cyclic subgroup of order 4 and a...Ch. 5 - In S3 , find elements and such that...Ch. 5 - Find group elements and in S5 such that...Ch. 5 - Represent the symmetry group of an equilateral...Ch. 5 - Prove that Sn is non-Abelian for all n3 .Ch. 5 - Prove that An is non-Abelian for all n4 .Ch. 5 - For n3 , let H=bSn(1)=1 or 2 and (2)=1or2 .Prove...Ch. 5 - Show that in S7 , the equation x2=(1234) has no...Ch. 5 - If (ab) and (cd) are distinct 2-cycles in Sn ,...Ch. 5 - Let and belong to Sn . Prove that 1 and are...Ch. 5 - Viewing the members of D4 as a group of...Ch. 5 - Viewing the members of D5 as a group of...Ch. 5 - Prob. 51ECh. 5 - Prob. 52ECh. 5 - Show that A5 has 24 elements of order 5, 20...Ch. 5 - Find a cyclic subgroup of A8 that has order 4....Ch. 5 - Prob. 55ECh. 5 - Prob. 56ECh. 5 - Show that every element in An for n3 can be...Ch. 5 - Show that for n3,Z(Sn)=[] .Ch. 5 - Prob. 59ECh. 5 - Use the Verhoeff check-digit scheme based on D5 to...Ch. 5 - Prob. 61ECh. 5 - (Indiana College Mathematics Competition) A...Ch. 5 - Prob. 63ECh. 5 - Find five subgroups of S5 of order 24.Ch. 5 - Why does the fact that the orders of the elements...Ch. 5 - Let a belong to Sn . Prove that divides n!Ch. 5 - Encrypt the message ATTACK POSTPONED using the...Ch. 5 - The message VAADENWCNHREDEYA was encrypted using...
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- Într-un bloc sunt apartamente cu 2 camere și apartamente cu 3 camere , în total 20 de apartamente și 45 de camere.Calculați câte apartamente sunt cu 2 camere și câte apartamente sunt cu 3 camere.arrow_forward1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.arrow_forwardQuestion 3 over a field K. In this question, MË(K) denotes the set of n × n matrices (a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is equivalent to A-¹? Justify your answer. (b) Let B be given by 8 B = 0 7 7 0 -7 7 Working over the field F2 with 2 elements, compute the rank of B as an element of M2(F2). (c) Let 1 C -1 1 [4] [6] and consider C as an element of M3(Q). Determine the minimal polynomial mc(x) and hence, or otherwise, show that C can not be diagonalised. [7] (d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write down all the eigenvalues. Show your working. [8]arrow_forward
- R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]arrow_forwardQuestion 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]arrow_forwardpart b pleasearrow_forward
- Question 5 (a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the matrix a cl |df equals 3 and the determinant of determinant of the matrix a+3b cl d+3e f ГЪ e [ c ] equals 2. Compute the [5] (b) Calculate the adjugate Adj (A) of the 2 × 2 matrix [1 2 A = over R. (c) Working over the field F3 with 3 elements, use row and column operations to put the matrix [6] 0123] A = 3210 into canonical form for equivalence and write down the canonical form. What is the rank of A as a matrix over F3? 4arrow_forwardQuestion 2 In this question, V = Q4 and - U = {(x, y, z, w) EV | x+y2w+ z = 0}, W = {(x, y, z, w) € V | x − 2y + w − z = 0}, Z = {(x, y, z, w) € V | xyzw = 0}. (a) Determine which of U, W, Z are subspaces of V. Justify your answers. (b) Show that UW is a subspace of V and determine its dimension. (c) Is VU+W? Is V = UW? Justify your answers. [10] [7] '00'arrow_forwardTools Sign in Different masses and Indicated velocities Rotational inert > C C Chegg 39. The balls shown have different masses and speeds. Rank the following from greatest to least: 2.0 m/s 8.5 m/s 9.0 m/s 12.0 m/s 1.0 kg A 1.2 kg B 0.8 kg C 5.0 kg D C a. The momenta b. The impulses needed to stop the balls Solved 39. The balls shown have different masses and speeds. | Chegg.com Images may be subject to copyright. Learn More Share H Save Visit > quizlet.com%2FBoyE3qwOAUqXvw95Fgh5Rw.jpg&imgrefurl=https%3A%2F%2Fquizlet.com%2F529359992%2Fc. Xarrow_forward
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