Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 4, Problem 6E
What do Exercises 3, 4, and 5 have in common? Try to make a generalization that includes these three cases.
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Chapter 4 Solutions
Contemporary Abstract Algebra
Ch. 4 - Find all generators of Z6,Z8,andZ20 .Ch. 4 - Suppose that a,b,andc are cyclic groups of orders...Ch. 4 - List the elements of the subgroups 20and10inZ30 ....Ch. 4 - List the elements of the subgroups 3and15inZ18 ....Ch. 4 - List the elements of the subgroups 3and7inU(20) .Ch. 4 - What do Exercises 3, 4, and 5 have in common? Try...Ch. 4 - Find an example of a noncyclic group, all of whose...Ch. 4 - Let a be an element of a group and let a=15 ....Ch. 4 - Prob. 9ECh. 4 - In Z24 , list all generators for the subgroup of...
Ch. 4 - Let G be a group and let aG . Prove that a1=a .Ch. 4 - In Z, find all generators of the subgroup 3 . If a...Ch. 4 - In Z24 , find a generator for 2110 . Suppose that...Ch. 4 - Suppose that a cyclic group G has exactly three...Ch. 4 - Let G be an Abelian group and let H=gG||g divides...Ch. 4 - Complete the statement: a|=|a2 if and only if |a|...Ch. 4 - Complete the statement: a2|=|a12 if and only if ....Ch. 4 - Let a be a group element and a= . Complete the...Ch. 4 - If a cyclic group has an element of infinite...Ch. 4 - Suppose that G is an Abelian group of order 35 and...Ch. 4 - Let G be a group and let a be an element of G. a....Ch. 4 - Prove that a group of order 3 must be cyclic.Ch. 4 - Let Z denote the group of integers under addition....Ch. 4 - For any element a in any group G, prove that a is...Ch. 4 - If d is a positive integer, d2 , and d divides n,...Ch. 4 - Find all generators of Z. Let a be a group element...Ch. 4 - Prove that C*, the group of nonzero complex...Ch. 4 - Let a be a group element that has infinite order....Ch. 4 - List all the elements of order 8 in Z8000000 . How...Ch. 4 - Suppose that G is a group with more than one...Ch. 4 - Let G be a finite group. Show that there exists a...Ch. 4 - Determine the subgroup lattice for Z12 ....Ch. 4 - Determine the subgroup lattice for Z8 . Generalize...Ch. 4 - Prove that a finite group is the union of proper...Ch. 4 - Show that the group of positive rational numbers...Ch. 4 - Consider the set {4, 8, 12, 16}. Show that this...Ch. 4 - Give an example of a group that has exactly 6...Ch. 4 - Let m and n be elements of the group Z. Find a...Ch. 4 - Suppose that a andb are group elements that...Ch. 4 - Prob. 40ECh. 4 - Prob. 41ECh. 4 - Let F and F’be distinct reflections in D21 . What...Ch. 4 - Suppose that H is a subgroup of a group G and H=10...Ch. 4 - Prob. 44ECh. 4 - If G is an infinite group, what can you say about...Ch. 4 - If G is a cyclic group of order n, prove that for...Ch. 4 - For each positive integer n, prove that C*, the...Ch. 4 - Prove or disprove that H=nZn is divisible by both...Ch. 4 - Prob. 49ECh. 4 - Prob. 50ECh. 4 - Prob. 51ECh. 4 - Prob. 52ECh. 4 - Prob. 53ECh. 4 - Prob. 54ECh. 4 - Prob. 55ECh. 4 - Prob. 56ECh. 4 - Prob. 57ECh. 4 - Prob. 58ECh. 4 - Prove that no group can have exactly two elements...Ch. 4 - Given the fact that U(49) is cyclic and has 42...Ch. 4 - Let a andb be elements of a group. If a=10andb=21...Ch. 4 - Let a andb belong to a group. If |a| and |b| are...Ch. 4 - Let a andb belong to a group. If a=24andb=10 ,...Ch. 4 - Prove that U(2n)(n3) is not cyclic.Ch. 4 - Prove that for any prime p and positive integer...Ch. 4 - Prove that Zn has an even number of generators if...Ch. 4 - If a5=12 , what are the possibilities for |a|? If...Ch. 4 - Suppose that x=n . Find a necessary and sufficient...Ch. 4 - Let a be a group element such that a=48 . For each...Ch. 4 - Prove that H={[1n01]|nZ} is a cyclic subgroup of...Ch. 4 - Suppose that |a| and |b| are elements of a group...Ch. 4 - Let a andb belong to a group. If a=12,b=22,andabe...Ch. 4 - Determine (81),(60)and(105) where is the Euler...Ch. 4 - If n is an even integer prove that (2n)=2(n) .Ch. 4 - Let a andb belong to some group. Suppose that...Ch. 4 - For every integer n greater than 2, prove that the...Ch. 4 - (2008 GRE Practice Exam) If x is an element of a...
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- Question 4 Find the value of the first element for the first row of the inverse matrix of matrix B. 3 Not yet answered B = Marked out of 5.00 · (³ ;) Flag question 7 [Provide your answer as an integer number (no fraction). For a decimal number, round your answer to 2 decimal places] Answer:arrow_forwardQuestion 2 Not yet answered Multiply the following Matrices together: [77-4 A = 36 Marked out of -5 -5 5.00 B = 3 5 Flag question -6 -7 ABarrow_forwardAssume {u1, U2, u3, u4} does not span R³. Select the best statement. A. {u1, U2, u3} spans R³ if u̸4 is a linear combination of other vectors in the set. B. We do not have sufficient information to determine whether {u₁, u2, u3} spans R³. C. {U1, U2, u3} spans R³ if u̸4 is a scalar multiple of another vector in the set. D. {u1, U2, u3} cannot span R³. E. {U1, U2, u3} spans R³ if u̸4 is the zero vector. F. none of the abovearrow_forward
- Select the best statement. A. If a set of vectors includes the zero vector 0, then the set of vectors can span R^ as long as the other vectors are distinct. n B. If a set of vectors includes the zero vector 0, then the set of vectors spans R precisely when the set with 0 excluded spans Rª. ○ C. If a set of vectors includes the zero vector 0, then the set of vectors can span Rn as long as it contains n vectors. ○ D. If a set of vectors includes the zero vector 0, then there is no reasonable way to determine if the set of vectors spans Rn. E. If a set of vectors includes the zero vector 0, then the set of vectors cannot span Rn. F. none of the abovearrow_forwardWhich of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.) ☐ A. { 7 4 3 13 -9 8 -17 7 ☐ B. 0 -8 3 ☐ C. 0 ☐ D. -5 ☐ E. 3 ☐ F. 4 THarrow_forward3 and = 5 3 ---8--8--8 Let = 3 U2 = 1 Select all of the vectors that are in the span of {u₁, u2, u3}. (Check every statement that is correct.) 3 ☐ A. The vector 3 is in the span. -1 3 ☐ B. The vector -5 75°1 is in the span. ГОЛ ☐ C. The vector 0 is in the span. 3 -4 is in the span. OD. The vector 0 3 ☐ E. All vectors in R³ are in the span. 3 F. The vector 9 -4 5 3 is in the span. 0 ☐ G. We cannot tell which vectors are i the span.arrow_forward
- (20 p) 1. Find a particular solution satisfying the given initial conditions for the third-order homogeneous linear equation given below. (See Section 5.2 in your textbook if you need a review of the subject.) y(3)+2y"-y-2y = 0; y(0) = 1, y'(0) = 2, y"(0) = 0; y₁ = e*, y2 = e¯x, y3 = e−2x (20 p) 2. Find a particular solution satisfying the given initial conditions for the second-order nonhomogeneous linear equation given below. (See Section 5.2 in your textbook if you need a review of the subject.) y"-2y-3y = 6; y(0) = 3, y'(0) = 11 yc = c₁ex + c2e³x; yp = −2 (60 p) 3. Find the general, and if possible, particular solutions of the linear systems of differential equations given below using the eigenvalue-eigenvector method. (See Section 7.3 in your textbook if you need a review of the subject.) = a) x 4x1 + x2, x2 = 6x1-x2 b) x=6x17x2, x2 = x1-2x2 c) x = 9x1+5x2, x2 = −6x1-2x2; x1(0) = 1, x2(0)=0arrow_forwardFind the perimeter and areaarrow_forwardAssume {u1, U2, us} spans R³. Select the best statement. A. {U1, U2, us, u4} spans R³ unless u is the zero vector. B. {U1, U2, us, u4} always spans R³. C. {U1, U2, us, u4} spans R³ unless u is a scalar multiple of another vector in the set. D. We do not have sufficient information to determine if {u₁, u2, 43, 114} spans R³. OE. {U1, U2, 3, 4} never spans R³. F. none of the abovearrow_forward
- Assume {u1, U2, 13, 14} spans R³. Select the best statement. A. {U1, U2, u3} never spans R³ since it is a proper subset of a spanning set. B. {U1, U2, u3} spans R³ unless one of the vectors is the zero vector. C. {u1, U2, us} spans R³ unless one of the vectors is a scalar multiple of another vector in the set. D. {U1, U2, us} always spans R³. E. {U1, U2, u3} may, but does not have to, span R³. F. none of the abovearrow_forwardLet H = span {u, v}. For each of the following sets of vectors determine whether H is a line or a plane. Select an Answer u = 3 1. -10 8-8 -2 ,v= 5 Select an Answer -2 u = 3 4 2. + 9 ,v= 6arrow_forwardSolve for the matrix X: X (2 7³) x + ( 2 ) - (112) 6 14 8arrow_forward
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