ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 9780357671139
Author: Gilbert
Publisher: CENGAGE L
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Chapter 4.5, Problem 11E
Find all normal subgroups of the alternating group
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The Course Name Real Analysis please Solve questions by Real Analysis
part 3 of the question is:
A power outage occurs 6 min after the ride started. Passengers must wait for their cage to be manually cranked into the lowest position in order to exit the ride. Sine function model: where h is the height of the last passenger above the ground measured in feet and t is the time of operation of the ride in minutes.
What is the height of the last passenger at the moment of the power outage? Verify your answer by evaluating the sine function model.
Will the last passenger to board the ride need to wait in order to exit the ride? Explain.
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
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- 2. The duration of the ride is 15 min. (a) How many times does the last passenger who boarded the ride make a complete loop on the Ferris wheel? (b) What is the position of that passenger when the ride ends?arrow_forward3. A scientist recorded the movement of a pendulum for 10 s. The scientist began recording when the pendulum was at its resting position. The pendulum then moved right (positive displacement) and left (negative displacement) several times. The pendulum took 4 s to swing to the right and the left and then return to its resting position. The pendulum's furthest distance to either side was 6 in. Graph the function that represents the pendulum's displacement as a function of time. Answer: f(t) (a) Write an equation to represent the displacement of the pendulum as a function of time. (b) Graph the function. 10 9 8 7 6 5 4 3 2 1 0 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -1 -5. -6 -7 -8 -9 -10-arrow_forwardA power outage occurs 6 min after the ride started. Passengers must wait for their cage to be manually cranked into the lowest position in order to exit the ride. Sine function model: h = −82.5 cos (3πt) + 97.5 where h is the height of the last passenger above the ground measured in feet and t is the time of operation of the ride in minutes. (a) What is the height of the last passenger at the moment of the power outage? Verify your answer by evaluating the sine function model. (b) Will the last passenger to board the ride need to wait in order to exit the ride? Explain.arrow_forward
- The Colossus Ferris wheel debuted at the 1984 New Orleans World's Fair. The ride is 180 ft tall, and passengers board the ride at an initial height of 15 ft above the ground. The height above ground, h, of a passenger on the ride is a periodic function of time, t. The graph displays the height above ground of the last passenger to board over the course of the 15 min ride. Height of Passenger in Ferris Wheel 180 160 140- €120 Height, h (ft) 100 80 60 40 20 0 ך 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time of operation, t (min) Sine function model: h = −82.5 cos (3πt) + 97.5 where h is the height of the passenger above the ground measured in feet and t is the time of operation of the ride in minutes. What is the period of the sine function model? Interpret the period you found in the context of the operation of the Ferris wheel. Answer:arrow_forward1. Graph the function f(x)=sin(x) −2¸ Answer: y -2π 一元 1 −1 -2 -3 -4+ 元 2πarrow_forward3. Graph the function f(x) = −(x-2)²+4 Answer: f(x) 6 5 4 3 2+ 1 -6-5 -4-3-2-1 × 1 2 3 4 5 6 -1 -2+ ရာ -3+ -4+ -5 -6arrow_forward
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