Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 4.6, Problem 10E
Find all homomorphic images of
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
How many different rectangles can be made whose side lengths, in centimeters, are counting numbers and whose are is 1,159 square centimeters? Draw and label all possible rectangles.
Co Given
show that
Solution
Take home
Су-15
1994
+19
09/2
4
=a
log
суто
-
1092
ж
= a-1
2+1+8
AI | SHOT ON S4
INFINIX CAMERA
a
Question 7. If det d e f
ghi
V3
= 2. Find det
-1
2
Question 8. Let A = 1
4
5
0
3
2.
1 Find adj (A)
2 Find det (A)
3
Find A-1
2g 2h 2i
-e-f
-d
273
2a 2b 2c
Chapter 4 Solutions
Elements Of Modern Algebra
Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False
Label each of the following...
Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - True or False
Label each of the following...Ch. 4.1 - Exercises
1. Express each permutation as a product...Ch. 4.1 - Exercises
2. Express each permutation as a product...Ch. 4.1 - Exercises
3. In each part of Exercise , decide...Ch. 4.1 - In each part of Exercise 2, decide whether the...Ch. 4.1 - Find the order of each permutation in Exercise 1....Ch. 4.1 - Exercises
6. Find the order of each permutation in...Ch. 4.1 - Exercises
7. Express each permutation in Exercise ...Ch. 4.1 - Express each permutation in Exercise 2 as a...Ch. 4.1 - Compute f2, f3, and f1 for each of the following...Ch. 4.1 - Let f=(1,2,3)(4,5). Compute each of the following...Ch. 4.1 - Exercises Let f=(1,6)(2,3,5,4). Compute each of...Ch. 4.1 - Exercises
12. Compute , the conjugate of by , for...Ch. 4.1 - Exercises
13. For the given permutations, and ,...Ch. 4.1 - Exercises
14. Write the permutation as a product...Ch. 4.1 - Exercises
15. Write the permutation as a product...Ch. 4.1 - Exercises List all the elements of the alternating...Ch. 4.1 - Exercises List all the elements of S4, written in...Ch. 4.1 - Exercises
18. Find all the distinct cyclic...Ch. 4.1 - Exercises
19. Find cyclic subgroups of that have...Ch. 4.1 - Exercises Construct a multiplication table for the...Ch. 4.1 - Exercises
21. Find all the distinct cyclic...Ch. 4.1 - Exercises Find an isomorphism from the octic group...Ch. 4.1 - Prob. 23ECh. 4.1 - Exercises In Section 3.3, the centralizer of an...Ch. 4.1 - Prob. 25ECh. 4.1 - Prob. 26ECh. 4.1 - Prob. 27ECh. 4.1 - Prob. 28ECh. 4.1 - Prob. 29ECh. 4.1 - Exercises Let be the mapping from Sn to the...Ch. 4.1 - Exercises Let f and g be disjoint cycles in Sn....Ch. 4.1 - Exercises Prove that the order of An is n!2.Ch. 4.1 - Exercises
33. Prove Theorem : Let be a...Ch. 4.2 - True or False
Label the following statements as...Ch. 4.2 - In Exercises 1- 9, let G be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let G be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let G be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let G be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let G be the given group. Write...Ch. 4.2 - In Exercises 1- 9, let be the given group. Write...Ch. 4.2 - 10. For each in the group, define a mapping by ...Ch. 4.2 - 11. For each in the group, define a mapping by ...Ch. 4.2 - Find the right regular representation of G as...Ch. 4.2 - For each a in the group G define a mapping ma:GG...Ch. 4.3 - Prob. 1TFECh. 4.3 - Prob. 2TFECh. 4.3 - Prob. 3TFECh. 4.3 - Prob. 4TFECh. 4.3 - True or False
Label each of the following...Ch. 4.3 - Prob. 6TFECh. 4.3 - The alternating group A4 on 4 elements is the same...Ch. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.3 - Prob. 10ECh. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Prob. 13ECh. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.3 - Prob. 16ECh. 4.3 - Prob. 17ECh. 4.3 - Prob. 18ECh. 4.3 - Prob. 19ECh. 4.3 - Prob. 20ECh. 4.3 - Prob. 21ECh. 4.3 - Prob. 22ECh. 4.3 - Construct a multiplication table for the group G...Ch. 4.3 - Prob. 24ECh. 4.3 - Construct a multiplication table for the group D5...Ch. 4.3 - List the elements of the group of rigid motions...Ch. 4.3 - Let G be the group of rigid motions of a cube....Ch. 4.3 - Let G be the group of rigid motions of a regular...Ch. 4.3 - Prob. 29ECh. 4.4 - True or False Label each of the following...Ch. 4.4 - True or False
Label each of the following...Ch. 4.4 - True or False Label each of the following...Ch. 4.4 - True or False
Label each of the following...Ch. 4.4 - True or False Label each of the following...Ch. 4.4 - True or False
Label each of the following...Ch. 4.4 - True or False Label each of the following...Ch. 4.4 - True or False
Label each of the following...Ch. 4.4 - 1. Consider , the groups of units in under...Ch. 4.4 - For each of the following subgroups H of the...Ch. 4.4 - In Exercises 3 and 4, let G be the octic group...Ch. 4.4 - In Exercises 3 and 4, let be the octic group in...Ch. 4.4 - Let H be the subgroup (1),(1,2) of S3. Find the...Ch. 4.4 - Let be the subgroup of .
Find the distinct left...Ch. 4.4 - In Exercises 7 and 8, let be the multiplicative...Ch. 4.4 - Prob. 8ECh. 4.4 - Let be a subgroup of a group with . Prove that ...Ch. 4.4 - Let be a subgroup of a group with . Prove that ...Ch. 4.4 - Let be a group of order 24. If is a subgroup of...Ch. 4.4 - Let H and K be subgroups of a group G and K a...Ch. 4.4 - Let H be a subgroup of the group G. Prove that if...Ch. 4.4 - Let H be a subgroup of a group G. Prove that gHg1...Ch. 4.4 - Prob. 15ECh. 4.4 - Let H be a subgroup of the group G. Prove that the...Ch. 4.4 - Show that a group of order 4 either is cyclic or...Ch. 4.4 - Let G be a group of finite order n. Prove that...Ch. 4.4 - Find the order of each of the following elements...Ch. 4.4 - Find all subgroups of the octic group D4.Ch. 4.4 - Prob. 21ECh. 4.4 - Lagranges Theorem states that the order of a...Ch. 4.4 - Find all subgroups of the quaternion group.Ch. 4.4 - Find two groups of order 6 that are not...Ch. 4.4 - If H and K are arbitrary subgroups of G, prove...Ch. 4.4 - Let p be prime and G the multiplicative group of...Ch. 4.4 - Prove that any group with prime order is cyclic.Ch. 4.4 - Let G be a group of order pq, where p and q are...Ch. 4.4 - Let be a group of order , where and are...Ch. 4.4 - Let G be an abelian group of order 2n, where n is...Ch. 4.4 - A subgroup H of the group Sn is called transitive...Ch. 4.4 - (See Exercise 31.) Suppose G is a group that is...Ch. 4.5 - True or False Label each of the following...Ch. 4.5 - Prob. 2TFECh. 4.5 - True or False
Label each of the following...Ch. 4.5 - True or False Label each of the following...Ch. 4.5 - True or False Label each of the following...Ch. 4.5 - True or False
Label each of the following...Ch. 4.5 - Prob. 7TFECh. 4.5 - Let G be the group and H the subgroup given in...Ch. 4.5 - 2. Show that is a normal subgroup of the...Ch. 4.5 - Prove or disprove that H={ [ 1a01 ]|a } is a...Ch. 4.5 - 4. Prove that the special linear group is a normal...Ch. 4.5 - 5. For any subgroup of the group , let denote the...Ch. 4.5 - Let H be a normal cyclic subgroup of a finite...Ch. 4.5 - Let H be a torsion subgroup of an abelian group G....Ch. 4.5 - Show that every subgroup of an abelian group is...Ch. 4.5 - 9. Consider the octic group of Example 3.
Find...Ch. 4.5 - 10. Find all normal subgroups of the octic...Ch. 4.5 - 11. Find all normal subgroups of the alternating...Ch. 4.5 - 12. Find all normal subgroups of the quaternion...Ch. 4.5 - Exercise 8 states that every subgroup of an...Ch. 4.5 - 14. Find groups and such that and the following...Ch. 4.5 - Find groups H and K such that the following...Ch. 4.5 - 16. Let be a subgroup of and assume that every...Ch. 4.5 - Prob. 17ECh. 4.5 - 18. If is a subgroup of , and is a normal...Ch. 4.5 -
19. With and as in Exercise 18, prove that is...Ch. 4.5 - Prob. 20ECh. 4.5 - With H and K as in Exercise 18, prove that K is a...Ch. 4.5 - 22. If and are both normal subgroups of , prove...Ch. 4.5 - 23. Prove that if and are normal subgroups of such...Ch. 4.5 - 24. The center of a group is defined as
...Ch. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - 27. Suppose is a normal subgroup of order of a...Ch. 4.5 - 28. For an arbitrary subgroup of the group , the...Ch. 4.5 - Find the normalizer of the subgroup (1),(1,3)(2,4)...Ch. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - Prob. 32ECh. 4.5 - Prob. 33ECh. 4.5 - Prob. 34ECh. 4.5 - Show that An has index 2 in Sn, and thereby...Ch. 4.5 - Prob. 36ECh. 4.5 - Prob. 37ECh. 4.5 - Let n be appositive integer, n1. Prove by...Ch. 4.5 - Prob. 39ECh. 4.5 - 40. Find the commutator subgroup of each of the...Ch. 4.6 - True or False Label each of the following...Ch. 4.6 - Prob. 2TFECh. 4.6 - True or False
Label each of the following...Ch. 4.6 - True or False
Label each of the following...Ch. 4.6 - True or False
Label each of the following...Ch. 4.6 - In Exercises , is a normal subgroup of the group...Ch. 4.6 - In Exercises , is a normal subgroup of the group...Ch. 4.6 - In Exercises , is a normal subgroup of the group...Ch. 4.6 - Prob. 4ECh. 4.6 - Prob. 5ECh. 4.6 - In Exercises , is a normal subgroup of the group...Ch. 4.6 - Let G be the multiplicative group of units U20...Ch. 4.6 - Suppose G1 and G2 are groups with normal subgroups...Ch. 4.6 - 9. Find all homomorphic images of the octic...Ch. 4.6 - 10. Find all homomorphic images of.
Ch. 4.6 - Find all homomorphic images of the quaternion...Ch. 4.6 - 12. Find all homomorphic images of each group in...Ch. 4.6 - Prob. 13ECh. 4.6 - Let G=I2,R,R2,R3,H,D,V,T be the multiplicative...Ch. 4.6 - 15. Repeat Exercise with, the multiplicative group...Ch. 4.6 - Prob. 16ECh. 4.6 - Prob. 17ECh. 4.6 - 18. If is a subgroup of the group such that for...Ch. 4.6 - Prob. 19ECh. 4.6 - Prob. 20ECh. 4.6 - Prob. 21ECh. 4.6 - Prob. 22ECh. 4.6 - Prob. 23ECh. 4.6 - 24. Let be a cyclic group. Prove that for every...Ch. 4.6 -
25. Prove or disprove that if a group has cyclic...Ch. 4.6 -
26. Prove or disprove that if a group has an...Ch. 4.6 -
27. a. Show that a cyclic group of order has a...Ch. 4.6 - Assume that is an epimorphism from the group G to...Ch. 4.6 -
29. Suppose is an epimorphism from the group to...Ch. 4.6 - Let G be a group with center Z(G)=C. Prove that if...Ch. 4.6 - 31. (See Exercise 30.) Prove that if and are...Ch. 4.6 - 32. Let be a fixed element of the group ....Ch. 4.6 - Prob. 33ECh. 4.6 - Prob. 34ECh. 4.6 - Prob. 35ECh. 4.6 - Prob. 36ECh. 4.6 - Let H and K be arbitrary groups and let HK denotes...Ch. 4.6 - Prob. 38ECh. 4.7 - True or False Label each of the following...Ch. 4.7 - Prob. 2TFECh. 4.7 - Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6...Ch. 4.7 - Prob. 2ECh. 4.7 - Prob. 3ECh. 4.7 - Prob. 4ECh. 4.7 - Prob. 5ECh. 4.7 - Prob. 6ECh. 4.7 - Write 20 as the direct sum of two of its...Ch. 4.7 - Prob. 8ECh. 4.7 - 9. Suppose that and are subgroups of the abelian...Ch. 4.7 - 10. Suppose that and are subgroups of the...Ch. 4.7 - 11. Assume that are subgroups of the abelian...Ch. 4.7 - Prob. 12ECh. 4.7 -
13. Assume that are subgroups of the abelian...Ch. 4.7 - 14. Let be an abelian group of order where and are...Ch. 4.7 - Let H1 and H2 be cyclic subgroups of the abelian...Ch. 4.7 - Prob. 16ECh. 4.7 - Prob. 17ECh. 4.7 - Prob. 18ECh. 4.7 - 19. a. Show that is isomorphic to , where the...Ch. 4.7 - Suppose that G and G are abelian groups such that...Ch. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - Prove that if r and s are relatively prime...Ch. 4.7 - Prob. 24ECh. 4.8 - True or False Label each of the following...Ch. 4.8 - Prob. 2TFECh. 4.8 - Prob. 3TFECh. 4.8 - Prob. 4TFECh. 4.8 - Prob. 5TFECh. 4.8 - Prob. 6TFECh. 4.8 - Prob. 1ECh. 4.8 - Prob. 2ECh. 4.8 - a. Find all Sylow 3-subgroups of the alternating...Ch. 4.8 - Find all Sylow 3-subgroups of the symmetric group...Ch. 4.8 - Prob. 5ECh. 4.8 - 6. For each of the following values of , describe...Ch. 4.8 - Let G be a group and gG. Prove that if H is a...Ch. 4.8 - Prob. 8ECh. 4.8 - 9. Determine which of the Sylow p-groups in each...Ch. 4.8 - Prob. 10ECh. 4.8 - 11. Show that is a generating set for the...Ch. 4.8 - Prob. 12ECh. 4.8 - If p1,p2,...,pr are distinct primes, prove that...Ch. 4.8 - Suppose that the abelian group G can be written as...Ch. 4.8 - 15. Assume that can be written as the direct sum...Ch. 4.8 - Prob. 16ECh. 4.8 - Prob. 17ECh. 4.8 - Prob. 18E
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- Question 1. Solve the system - x1 x2 + 3x3 + 2x4 -x1 + x22x3 + x4 2x12x2+7x3+7x4 Question 2. Consider the system = 1 =-2 = 1 3x1 - x2 + ax3 = 1 x1 + 3x2 + 2x3 x12x2+2x3 = -b = 4 1 For what values of a, b will the system be inconsistent? 2 For what values of a, b will the system have only one solution? For what values of a, b will the saystem have infinitely many solutions?arrow_forwardQuestion 5. Let A, B, C ben x n-matrices, S is nonsigular. If A = S-1 BS, show that det (A) = det (B) Question 6. For what values of k is the matrix A = (2- k -1 -1 2) singular? karrow_forward1 4 5 Question 3. Find A-1 (if exists), where A = -3 -1 -2 2 3 4 Question 4. State 4 equivalent conditions for a matrix A to be nonsingulararrow_forward
- How long is a guy wire reaching from the top of a 15-foot pole to a point on the ground 9-feet from the pole? Question content area bottom Part 1 The guy wire is exactly feet long. (Type an exact answer, using radicals as needed.) Part 2 The guy wire is approximatelyfeet long. (Round to the nearest thousandth.)arrow_forwardQuestion 6 Not yet answered Marked out of 5.00 Flag question = If (4,6,-11) and (-12,-16,4), = Compute the cross product vx w karrow_forwardConsider the following vector field v^-> (x,y): v^->(x,y)=2yi−xj What is the magnitude of the vector v⃗ located in point (13,9)? [Provide your answer as an integer number (no fraction). For a decimal number, round your answer to 2 decimal places]arrow_forward
- 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
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Algebra: Structure And Method, Book 1AlgebraISBN:9780395977224Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. ColePublisher:McDougal Littell
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Big Ideas Math A Bridge To Success Algebra 1: Stu...AlgebraISBN:9781680331141Author:HOUGHTON MIFFLIN HARCOURTPublisher:Houghton Mifflin Harcourt
Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Algebra: Structure And Method, Book 1
Algebra
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:McDougal Littell
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Big Ideas Math A Bridge To Success Algebra 1: Stu...
Algebra
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:Houghton Mifflin Harcourt
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Vector Spaces | Definition & Examples; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=72GtkP6nP_A;License: Standard YouTube License, CC-BY
Understanding Vector Spaces; Author: Professor Dave Explains;https://www.youtube.com/watch?v=EP2ghkO0lSk;License: Standard YouTube License, CC-BY