tutior const ০ 220 Groups 7. If is a homomorphism from G to H and o is a homomorphism from H to K, show that od is a homomorphism from G to K. How and Ker od related? If and o are onto and G is finite, are Ker describe [Ker od:Ker ] in terms of IHI and IKI. 8. Let G be a group of permutations. For each or in G, define +1 if o is an even permutation, if o is an odd permutation. sgn(o) Prove that sgn is a homomorphism from G to the multiplicative group +1, -1}. What is the kernel? Why does this homomor- phism allow you index 2? Why does this prove Exercise 23 of Chapter 5? 9. Prove that the mapping from G H to G given by (g, h)-g is a homomorphism. What is the kernel? This mapping is called the to conclude that A, is a normal subgroup of S of n projection of G H onto G. 10. Let G be a subgroup of some dihedral group. For each x in G, define 202guo +1 if x is a rotation, -1 if x is a reflection. Ф(х) a homomorphism from G to the multiplicative Prove that is group {+1, -1}. What is the kernel? Why does this prove Exercise 25 of Chapter 3? 11. Prove that (Z Z)/(((a, 0)) X ((0, b))) 12. Suppose that k is a divisor of n. Prove that Z/(k) Z 13. Prove that (A B)/(A {e}) ~ B. 14. Explain why the correspondence x3x from Z2 to Z0 is not a homomorphism. 15. Suppose that is a homomorphism from Z0 to Za0 and Ker (0, 10, 20). If d(23) = 9, determine all elements that map to 9. 16. Prove that there is no homomorphism from Zg Z, onto Z4 Z 17. Prove that there is no homomorphism from Z,16 Z, onto ZgZ 18. Can there be a homomorphism from Z, Z onto Z? Can there be a homomorphism from Z16 onto Z, Z,? Explain your answers. 19. Suppose that there is a homomorphism from Z1, and that is not one-to-one. Determine . 20. How many homomorphisms are there from Z onto Z? How many are there to Z? 21. If is a homomorphism from Z onto a group of order 5, deter- mine the kernel of d. isomorphic to Z®Z. is 10 11 30 to some group 20 www
tutior const ০ 220 Groups 7. If is a homomorphism from G to H and o is a homomorphism from H to K, show that od is a homomorphism from G to K. How and Ker od related? If and o are onto and G is finite, are Ker describe [Ker od:Ker ] in terms of IHI and IKI. 8. Let G be a group of permutations. For each or in G, define +1 if o is an even permutation, if o is an odd permutation. sgn(o) Prove that sgn is a homomorphism from G to the multiplicative group +1, -1}. What is the kernel? Why does this homomor- phism allow you index 2? Why does this prove Exercise 23 of Chapter 5? 9. Prove that the mapping from G H to G given by (g, h)-g is a homomorphism. What is the kernel? This mapping is called the to conclude that A, is a normal subgroup of S of n projection of G H onto G. 10. Let G be a subgroup of some dihedral group. For each x in G, define 202guo +1 if x is a rotation, -1 if x is a reflection. Ф(х) a homomorphism from G to the multiplicative Prove that is group {+1, -1}. What is the kernel? Why does this prove Exercise 25 of Chapter 3? 11. Prove that (Z Z)/(((a, 0)) X ((0, b))) 12. Suppose that k is a divisor of n. Prove that Z/(k) Z 13. Prove that (A B)/(A {e}) ~ B. 14. Explain why the correspondence x3x from Z2 to Z0 is not a homomorphism. 15. Suppose that is a homomorphism from Z0 to Za0 and Ker (0, 10, 20). If d(23) = 9, determine all elements that map to 9. 16. Prove that there is no homomorphism from Zg Z, onto Z4 Z 17. Prove that there is no homomorphism from Z,16 Z, onto ZgZ 18. Can there be a homomorphism from Z, Z onto Z? Can there be a homomorphism from Z16 onto Z, Z,? Explain your answers. 19. Suppose that there is a homomorphism from Z1, and that is not one-to-one. Determine . 20. How many homomorphisms are there from Z onto Z? How many are there to Z? 21. If is a homomorphism from Z onto a group of order 5, deter- mine the kernel of d. isomorphic to Z®Z. is 10 11 30 to some group 20 www
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Topic Video
Question
12
![tutior
const
০
220
Groups
7. If is a homomorphism from G to H and o is a homomorphism
from H to K, show that od is a homomorphism from G to K. How
and Ker od related? If and o are onto and G is finite,
are Ker
describe [Ker od:Ker ] in terms of IHI and IKI.
8. Let G be a group of permutations. For each or in G, define
+1 if o is an even permutation,
if o is an odd permutation.
sgn(o)
Prove that sgn is a homomorphism from G to the multiplicative
group +1, -1}. What is the kernel? Why does this homomor-
phism allow you
index 2? Why does this prove Exercise 23 of Chapter 5?
9. Prove that the mapping from G H to G given by (g, h)-g is a
homomorphism. What is the kernel? This mapping is called the
to conclude that A, is a normal subgroup of S of
n
projection of G H onto G.
10. Let G be a subgroup of some dihedral group. For each x in G, define
202guo
+1 if x is a rotation,
-1 if x is a reflection.
Ф(х)
a homomorphism from G to the multiplicative
Prove that is
group {+1, -1}. What is the kernel? Why does this prove Exercise
25 of Chapter 3?
11. Prove that (Z Z)/(((a, 0)) X ((0, b)))
12. Suppose that k is a divisor of n. Prove that Z/(k) Z
13. Prove that (A B)/(A {e}) ~ B.
14. Explain why the correspondence x3x from Z2 to Z0 is not a
homomorphism.
15. Suppose that is a homomorphism from Z0 to Za0 and Ker
(0, 10, 20). If d(23) = 9, determine all elements that map to 9.
16. Prove that there is no homomorphism from Zg Z, onto Z4 Z
17. Prove that there is no homomorphism from Z,16 Z, onto ZgZ
18. Can there be a homomorphism from Z, Z onto Z? Can there be
a homomorphism from Z16 onto Z, Z,? Explain your answers.
19. Suppose that there is a homomorphism from Z1,
and that is not one-to-one. Determine .
20. How many homomorphisms are there from Z onto Z? How many
are there to Z?
21. If is a homomorphism from Z onto a group of order 5, deter-
mine the kernel of d.
isomorphic to Z®Z.
is
10
11
30
to some group
20
www](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff82c0bab-f337-4781-985d-1029c419adb6%2Fb5b6ad82-39f6-4e32-a7b2-199273a7fdc1%2Fojh85zs.jpeg&w=3840&q=75)
Transcribed Image Text:tutior
const
০
220
Groups
7. If is a homomorphism from G to H and o is a homomorphism
from H to K, show that od is a homomorphism from G to K. How
and Ker od related? If and o are onto and G is finite,
are Ker
describe [Ker od:Ker ] in terms of IHI and IKI.
8. Let G be a group of permutations. For each or in G, define
+1 if o is an even permutation,
if o is an odd permutation.
sgn(o)
Prove that sgn is a homomorphism from G to the multiplicative
group +1, -1}. What is the kernel? Why does this homomor-
phism allow you
index 2? Why does this prove Exercise 23 of Chapter 5?
9. Prove that the mapping from G H to G given by (g, h)-g is a
homomorphism. What is the kernel? This mapping is called the
to conclude that A, is a normal subgroup of S of
n
projection of G H onto G.
10. Let G be a subgroup of some dihedral group. For each x in G, define
202guo
+1 if x is a rotation,
-1 if x is a reflection.
Ф(х)
a homomorphism from G to the multiplicative
Prove that is
group {+1, -1}. What is the kernel? Why does this prove Exercise
25 of Chapter 3?
11. Prove that (Z Z)/(((a, 0)) X ((0, b)))
12. Suppose that k is a divisor of n. Prove that Z/(k) Z
13. Prove that (A B)/(A {e}) ~ B.
14. Explain why the correspondence x3x from Z2 to Z0 is not a
homomorphism.
15. Suppose that is a homomorphism from Z0 to Za0 and Ker
(0, 10, 20). If d(23) = 9, determine all elements that map to 9.
16. Prove that there is no homomorphism from Zg Z, onto Z4 Z
17. Prove that there is no homomorphism from Z,16 Z, onto ZgZ
18. Can there be a homomorphism from Z, Z onto Z? Can there be
a homomorphism from Z16 onto Z, Z,? Explain your answers.
19. Suppose that there is a homomorphism from Z1,
and that is not one-to-one. Determine .
20. How many homomorphisms are there from Z onto Z? How many
are there to Z?
21. If is a homomorphism from Z onto a group of order 5, deter-
mine the kernel of d.
isomorphic to Z®Z.
is
10
11
30
to some group
20
www
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
