9 Normal Subgroups and Factor Groups 201 9. Prove that if H has index 2 in G, then H is normal in G. (This exer- cise is referred to in Chapters 24 and 25 and this chapter.) gg ct d 10. Let H {(1), (12)(34)} in A,. a. Show that H is not normal in A,. n b. Referring to the multiplication table for A, in Table 5.1 on page 111, show that, although a H = a,H and a,H = a,,H, it is not that agasH= a,a1 cosets of H do not form a group under coset multiplication. H.Explain why this proves that the left 11. Let G Z U(4), H = ((2, 3), and K = is not isomorphic to G/K. (This shows that H~ K does not imply that G/H GIK.) 12. Prove that a factor group of a cyclic group is cyclic. 13. Prove that a factor group ((2, 1)). Show that G/H of Abelian group is Abelian. 14. What is the order of the element 14 (8) in the factor an group Zn(8)? 15. What is the order of the element 4U(105) in the factor group U(105)/U,(105)? 16. Recall that Z(D) = RZ(D) in the factor group DdZ(D? {Rp, R). What is the order of the element 180 60 ZI20) and H = (4)/(20). List the elements of H and G/H. 17. Let G 18. What is the order of the factor group Z15)? 19. What is the order of the factor group (Z U(10))/(2, 9))? 20. Construct the Cayley table for U(20)/U(20). 60 10 21. Prove that an Abelian group of order 33 is cyclic. 22. Determine the order of (Z D Z)((2, 2). Is the group cyclic? Z)/((4, 2). Is the group cyclic? 23. Determine the order of (Z 24. The group (Z, Z2)(2, 2)) is isomorphic to one of Zg, Z Z, or Z, Z, Z,. Determine which one by elimination. 25. Let G U(32) and H (1, 31). The group GIH is isomorphic to one of Zg, Z Z, or Z, e Z, Z. Determine which one by elimination. 4 26. Let G be the group of quaternions given by the table in Exercise 4 of the Supplementary Exercises for Chapters 1-4, and let H be the subgroup {e, a2}. Is G/H isomorphic to Z, or Z, Z,? 27. Let G = U(16), H = {1, 15), and K {1, 9}. Are H and K iso- morphic? Are G/H and G/K isomorphic? 28. Let G Z, Z, H= {(0, 0), (2, 0), (0, 2), (2, 2)}, and K= {(1, 2)). Is GIH isomorphic to Z, or Z, Z,? Is GIK isomorphic to Z, or Z,Oz? 4 4 29. Prove that AOZ has no subgroup of order 18 N aull

Question

28

9
Normal Subgroups and Factor Groups
201
9. Prove that if H has index 2 in G, then H is normal in G. (This exer-
cise is referred to in Chapters 24 and 25 and this chapter.)
gg
ct
d
10. Let H {(1), (12)(34)} in A,.
a. Show that H is not normal in A,.
n
b. Referring to the multiplication table for A, in Table 5.1 on page
111, show that, although a H = a,H and a,H = a,,H, it is not
that agasH= a,a1
cosets of H do not form a group under coset multiplication.
H.Explain why this proves that the left
11. Let G Z U(4), H = ((2, 3), and K =
is not isomorphic to G/K. (This shows that H~ K does not imply
that G/H GIK.)
12. Prove that a factor group of a cyclic group is cyclic.
13. Prove that a factor group
((2, 1)). Show that G/H
of
Abelian group is Abelian.
14. What is the order of the element 14 (8) in the factor
an
group
Zn(8)?
15. What is the order of the element 4U(105) in the factor group
U(105)/U,(105)?
16. Recall that Z(D) =
RZ(D) in the factor group DdZ(D?
{Rp, R). What is the order of the element
180
60
ZI20) and H = (4)/(20). List the elements of H and G/H.
17. Let G
18. What is the order of the factor group Z15)?
19. What is the order of the factor group (Z U(10))/(2, 9))?
20. Construct the Cayley table for U(20)/U(20).
60
10
21. Prove that an Abelian group of order 33 is cyclic.
22. Determine the order of (Z D Z)((2, 2). Is the group cyclic?
Z)/((4, 2). Is the group cyclic?
23. Determine the order of (Z
24. The group (Z, Z2)(2, 2)) is isomorphic to one of Zg, Z Z, or
Z, Z, Z,. Determine which one by elimination.
25. Let G U(32) and H (1, 31). The group GIH is isomorphic to
one of Zg, Z Z, or Z, e Z, Z. Determine which one by
elimination.
4
26. Let G be the group of quaternions given by the table in Exercise 4
of the Supplementary Exercises for Chapters 1-4, and let H be the
subgroup {e, a2}. Is G/H isomorphic to Z, or Z, Z,?
27. Let G = U(16), H = {1, 15), and K {1, 9}. Are H and K iso-
morphic? Are G/H and G/K isomorphic?
28. Let G Z, Z, H= {(0, 0), (2, 0), (0, 2), (2, 2)}, and K= {(1, 2)).
Is GIH isomorphic to Z, or Z, Z,? Is GIK isomorphic to Z, or
Z,Oz?
4
4
29. Prove that AOZ has no subgroup of order 18
N
aull
Expand
Transcribed Image Text

9 Normal Subgroups and Factor Groups 201 9. Prove that if H has index 2 in G, then H is normal in G. (This exer- cise is referred to in Chapters 24 and 25 and this chapter.) gg ct d 10. Let H {(1), (12)(34)} in A,. a. Show that H is not normal in A,. n b. Referring to the multiplication table for A, in Table 5.1 on page 111, show that, although a H = a,H and a,H = a,,H, it is not that agasH= a,a1 cosets of H do not form a group under coset multiplication. H.Explain why this proves that the left 11. Let G Z U(4), H = ((2, 3), and K = is not isomorphic to G/K. (This shows that H~ K does not imply that G/H GIK.) 12. Prove that a factor group of a cyclic group is cyclic. 13. Prove that a factor group ((2, 1)). Show that G/H of Abelian group is Abelian. 14. What is the order of the element 14 (8) in the factor an group Zn(8)? 15. What is the order of the element 4U(105) in the factor group U(105)/U,(105)? 16. Recall that Z(D) = RZ(D) in the factor group DdZ(D? {Rp, R). What is the order of the element 180 60 ZI20) and H = (4)/(20). List the elements of H and G/H. 17. Let G 18. What is the order of the factor group Z15)? 19. What is the order of the factor group (Z U(10))/(2, 9))? 20. Construct the Cayley table for U(20)/U(20). 60 10 21. Prove that an Abelian group of order 33 is cyclic. 22. Determine the order of (Z D Z)((2, 2). Is the group cyclic? Z)/((4, 2). Is the group cyclic? 23. Determine the order of (Z 24. The group (Z, Z2)(2, 2)) is isomorphic to one of Zg, Z Z, or Z, Z, Z,. Determine which one by elimination. 25. Let G U(32) and H (1, 31). The group GIH is isomorphic to one of Zg, Z Z, or Z, e Z, Z. Determine which one by elimination. 4 26. Let G be the group of quaternions given by the table in Exercise 4 of the Supplementary Exercises for Chapters 1-4, and let H be the subgroup {e, a2}. Is G/H isomorphic to Z, or Z, Z,? 27. Let G = U(16), H = {1, 15), and K {1, 9}. Are H and K iso- morphic? Are G/H and G/K isomorphic? 28. Let G Z, Z, H= {(0, 0), (2, 0), (0, 2), (2, 2)}, and K= {(1, 2)). Is GIH isomorphic to Z, or Z, Z,? Is GIK isomorphic to Z, or Z,Oz? 4 4 29. Prove that AOZ has no subgroup of order 18 N aull

Expert Answer

Want to see the step-by-step answer?

Check out a sample Q&A here.

Want to see this answer and more?

Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes!*

*Response times may vary by subject and question complexity. Median response time is 34 minutes for paid subscribers and may be longer for promotional offers.
Tagged in
Math
Advanced Math

Related Advanced Math Q&A

Find answers to questions asked by students like you.

Q: Using either logarithms or a graphing calculator, find the time required for the initial amount to b...

A: To work out the time taken for the compounded amout to reach the given figure under the given condit...

Q: (b) Let f V -» V be any linear map of vector spaces over a field K. Recall that, for any polynomial ...

A: As per norms, three parts are answered. To prove the required properties of polynomials of a linear ...

Q: Use a double integral in polar coordinates to evaluate x dA where D is the region inside r 1- sin an...

A: To compute the required double integral using polar coordinates

Q: How do I find the upper bound of this series?

A: To find a constant M such that the sum of the given infinite series is less than or equal to M

Q: 54

A: The problem concerns the groujp properties of the Quaternion group of order 8,

Q: X Document1 - Word AutoSave evie tucker Off ET Design References Mailings Review View Help Search Sh...

A: The inverse of the matrix A is given by the formula,

Q: A die is loaded, and it was determined that the probability distribution associated with the experim...

A: (a)Probability of the number being even = sum of the probabilities of all even sample events.P(2) +P...

Q: [3 2 Find all the eigenvalues for A = Find one eigenvector corresponding to each eigenvalue 1 4

A: Consider the given matrix as:

Q: How much money should be deposited today in an account that earns 5% compounded semiannually so that...

A: To work out the initial amount  to be invested that yields the given amount .

Q: You want to be able to withdraw $40,000 each year for 15 years. Your account earns 6% interest.a) Ho...

A: From the given information, the amounts withdraw $40, 000.The number of years is 15 and interest is ...

Q: In Exercises 18-30 find a fundamental set of solutions, given that yı is a solution. х2у" — 2ху' + (...

A: Click to see the answer

Q: Estimate the amount of carbon dioxide released in burning one gallon of gasoline. Give your answer i...

A: Obtain the amount of gasoline and carbon needed as follows.

Q: 3 Estimate e2* sin(x) dx 1 using the Trapezoidal Rule with n = 1,2,4 and 8 segments 2) using Romberg...

A: Part (1): (a)The trapezoidal rule stats that: 

Q: Let (X, d) be a metric space. a) Assume that A is a (finite or infinite) collection of open sets. Sh...

A: To orove the required topological properties of open sets in a metric space (X,d)

Q: Draw an Eulee Diagram

A: The given premises are:

Q: For the given function, find all values of ?c over the interval (1/2,20) stipulated by Rolle's Theor...

A: To solve for the values c for the given function and the interval , as per Rolle's theorem

Q: In Exercises 1-14 find a particular solution. у" — 2у — Зу — е*(-8 + 3х) 3.

A: We need to find  the particular solution only :

Q: Solve the following system of linear equations:   Which one of the following statements best describ...

A: Consider the given system of linear equations.

Q: Find the simple interest. Assume a 360-day year. Round results to the nearest cent. 1) $3180 at 5% f...

A: Recall the following fact.

Q: Consider the following determinant. 1 6 7 8 (a) Evaluate the determinant. (b) Use the determinant to...

A: To calcuate the value of the determinant, and hence to conclude whether the corresponding matrix is ...

Q: For each signal below, compute its exponential Fourier series. (а) 8 4 12 8 (a) (b) -12 8 8 12

A: a) From the given figure, the periodic waveform shown in fig with a period T= 8 can be expressed as

Q: Prove that ( n2 + 1) / (n + 1) diverges to +∞

A: Let

Q: Use inverse matrices to find the solution of the system of equations. y 2z = 9 у + 2х + 2y + х+ 2х +...

A: To find the values of x, y and z by inverting the matrix of the given system of equations

Q: QUESTION IS ATTACHED  (PLEASE DO PART B ONLY I HAVE COMPLETED PART A ?)

A: 1 B) to show that the particular APs A(1,0) and A(0,1) form a basis for the 2-dimensional vector spa...

Q: 2) Which of the following are linear transformations from R2 - R2? (a) Shear: T(x, y) (x + cy,y) for...

A: As per norms, the first three questions ( of the multiple pars problem ) are answered. To identify t...

Q: In a group of 50 ballpoint pens on a shelf in the stationary department of Metro Department Store, 2...

A: (a) There are total 50 pens so, N = 50 out if 50, 2 pens are defective and else are not defective. L...

Q: For the given function, find all values of c over the interval (4, 8) stipulated by Rolle's Theorem ...

A: To solve for all the values c, where the derivative vanishes.

Q: 1 2.5 4.4 2 13.6 4 20.7 25.6 36.2 Using techniques in nonlinear regression, determine the constants ...

A: Consider the given table.The given table is as follows. 

Q: X Document1 - Word AutoSave evie tucker Off ET Mailings Review Share File Design References View Hel...

A: Given column vectors are,

Q: of points of S. 6.30 Show that a set consisting of a finite number of points in E is bounded. Gnita ...

A: Given, a set consisting of a finite number of points in E.

Q: Nedd help with d)

A: (d) To determine if the given mapping T is a linear transformation on the plane R^2

Q: Use inverse matrices to solve the system of linear equations. 2y Зх + Зу 5x = 6 12 (х, у) 3D

A: To find the values of x and y satisfying the system using matrix inverse

Q: According to a census​ bureau, from 2000 to 2016 the population of a certain region grew from 261 mi...

A: It is given that the population of a certain region grew from 261 million to 301 million.

Q: Use the Laplace transform to solve the given initial-value problem. y''+2y'+y=0,    y(0)=1,    y'(0)...

A: Click to see the answer

Q: The initial tableau of a linear programming problem is given. Use the simplex method to solve it. X-...

A: Given, the simplex table of the linear programming problem is

Q: c(x)=3x4-2x3-21x2-4x+12 What are the zeros of c(x)?

A: To find all the solutions of c(x)=0

Q: X Document1 - Word AutoSave evie tucker Off ET Design References Mailings Review View Help Search Sh...

A: If there exists a Non-zero vector X such that AX = λX then X is called Eigen vector of matrix A corr...

Q: What is the Value of A?

A: Consider the given matrices,

Q: 3. In the poker game, a Poker hand of five cards is dealt to the player. The order of the five cards...

A: Obtain the number of poker hand possible as follows.The number of different possible poker hands is ...

Q: An open tank with a 200L capacity is half-filled with pure water. A salt solution with a concentrati...

A: Let’s set up a differential equation. We are given r1 = 15L/h, r2 = 10L/h, c1 = 10g/L, V(0) = 100L  ...

Q: m(x)=x4-2x2-24 What are the zeros of m(x)?

A: To find all the values of x for which m(x)=0 (zero of a polynomial m(x) is just a root of m(x)=0)