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
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 32E: (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup...
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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
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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
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Elements Of Modern Algebra
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Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,