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

24

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

A: Given column vectors are,

Q: 8.

A: Problem concerns quotients G/H , where G and H are subgroups of Z =the additiv group of integers

Q: Find a way to calculate accurate values of f(x) for small values of x.f(x) =[sqrt(1 + x^2) - 1] / x^...

A: Given function is:

Q: last two subparts

A: To determine the validity of 2d) and 2e)  

Q: (b) Let v, 2,... , vn be a basis of a vector space V. Suppose f: V -> V is a lincar map so that Vi -...

A: To work out the required properties of the linear operator f.

Q: Compute the following for the function defined below. Simplify your answers. Note: Use lowercase let...

A: Click to see the answer

Q: Let A = [1,2,3,4; 0,0,1,-1; 2,0,0,-3;1,2,2,5] Determine the rank of A.

A: Consider the given matrix A,

Q: Can you help me fix this problem thank you function of complex variable class

A: Given:

Q: C 4. Sketch the solid whose volume is given by the iterated integral. (a) » 3π{2 π{2 » 4 0 » 2 1 r ...

A: To identify the solid defined by the (three) limits of integration

Q: Let z=x+iy  Let w=a+ib

A: To prove the equality of the two given angles

Q: Evaluate the double integral of 6y2  The Domian is the triangular region with vertices (0, 1), (1, ...

A: Consider the function,

Q: Show that the rank of the linear mapping 7.32. Bar Y3= 301 +302 y = 201 +20203, 31-02-203 1 V2 is eq...

A: To prove that the rank of the matrix (representing the given linear transformation ) is 2 and to det...

Q: very logical, easy to follow and detailed solution, please!

A: Given that V and W be vector spaces over a field K and f is any linear map that is

Q: How can you draw a Euler diagram to show the following is invalid

A: P implies q The sentence p implies q is true if and only if the p is false or q is true or (the sent...

Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: Advanced Math Question

A: Let E is the region between z = (x2+y2-a2)1/2 and z = (x2+y2-a2)1/2 over the domain D on the xy plan...

Q: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,...

A: To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in ...

Q: Find the length of the curve. r(t) 4 it2j3k, 0 st s 1

A: Consider the curve x = 4, y = t2 and z = t3.The derivatives are

Q: #3 a and b

A: Click to see the answer

Q: Hi the question is attached

A: To  describe the locus of all z satisfying the given conditions

Q: 4 True or False. Explain briefly (you can use diagram to show, or one or two short sentences). (a) F...

A: Click to see the answer

Q: A fair die is cast three times. What is the probability that it shows an even number on the first to...

A: According to the given question:Outcomes when a die is rolled are: 1,2,3,4,5,6In the given question ...

Q: How do I solve the simultaneous following equations using matrix algebra?x + y + z = 3 x + 2y - z = ...

A: To apply matrix method to solve the given simultaneous equations

Q: Exercise 2: Hard] Let f,g be functions such that limc f(x) = L and lim,c g(x) = L (a) Let h(x) max{f...

A: Part (A): Let f and g are functions and satisfied the given conditions: Now max is defined as:

Q: Find the interest rate for each deposit and compound amount. 6) $8000 accumulating to $12,384.48, co...

A: Recall the following fact.

Q: Consider the following quadratic function f(x) -7x2 + 12x - 11 Find the vertex - | (x, y) Determine ...

A: To locate the vertex and decide whether it corresponds to a maximum or minimum

Q: linear algebra

A: Find the eigenvalue of the given system as follows.

Q: Find the maturity value and the amount of simple interest earned. Round to the nearest cent. 2) $548...

A: Recall the following fact.

Q: f(x)=3x3 + 23x2 - 22x - 112 Factor f(x) given that -2 is a zero

A: Given that –2 is a zero, thus (x–(–2)) is a factor of the given function.

Q: 1 2.5 4.4 2 13.6 4 20.7 4 25.6 36.2 Calculate and interpret the coefficient of determination for the...

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

Q: In Exercises 1-17 find a particular solution. y3 1. 7 cos x - sinx

A: The given equation

Q: Write f(x) |6x + 7 as a piece-wise defined function. if x 2 f(x) if x <

A: Consider the given function,f(x) = |6x + 7|Equate f(x) = 0 and find the value of x,

Q: Advanced Math Question

A: To determine the objective function and the constraints in the word problem

Q: How can I prove by induction that 2^n + 1 is divisible by three for all odd integers

A: To apply mathematical induction to show that 2^n+1 is divisible by 3 if n is odd

Q: Jill Sharp, a fitness trainer, has an exercise regimen that includes running, swimming, and walking....

A: According to the given information:Let, r be the number of hours jill spend on running s be the numb...

Q: Linear Alegbra

A: To find the inverse of a matrix, augment the given matrix with identity matrix and apply row operati...

Q: Find the eigenvalues and number of independent eigenvectors.  (Hint:  1 is an eigenvalue.)

A: Consider given matrix as,

Q: Palindrome numbers are numbers that read the same forwards or backward. For example, 55 and 94549 ar...

A: TO count the number of palindromic numbers in the given range.

Q: A polygon is said to be convex if any line segment connecting two vertices of the polygon lies entir...

A: We need to prove that the sum of the interior angles in a convex polygon with n vertices is (n-2)π b...

Q: In Exercises 22-26 solve the initial value problem. =-ex(6 cos x 4 sin x), y(0) 1, y'(0)= 4 y"-2y' 2...

A: We need to solve the initial value problem,

Q: I need help with 12.3.3.please.

A: To find the perimeter of the given lemniscate curve