ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 9780357671139
Author: Gilbert
Publisher: CENGAGE L
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 4.1, Problem 19E
Exercises
Find cyclic subgroups of
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
13) Let U = {j, k, l, m, n, o, p} be the universal set. Let V = {m, o,p), W = {l,o, k}, and X = {j,k). List the elements of
the following sets and the cardinal number of each set.
a) W° and n(W)
b) (VUW) and n((V U W)')
c) VUWUX and n(V U W UX)
d) vnWnX and n(V WnX)
9) Use the Venn Diagram given below to determine the number elements in each of the following sets.
a) n(A).
b) n(A° UBC).
U
B
oh
a
k
gy
ท
W
z r
e t
་
C
10) Find n(K) given that n(T) = 7,n(KT) = 5,n(KUT) = 13.
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
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- 7) Use the Venn Diagram below to determine the sets A, B, and U. A = B = U = Blue Orange white Yellow Black Pink Purple green Grey brown Uarrow_forward8) Use the Venn diagram provided to shade An Bº. A B U Darrow_forward5) Describe the difference between disjoint sets and overlapping sets.arrow_forward
- 12) Suppose U = {a,b,c,d,e) and A = {a, b, c, e) and B = (c,d,e). Determine (An B).arrow_forward1) Use the roster method to list the elements of the set consisting of: a) All positive multiples of 3 that are less than 20. b) Nothing (An empty set).arrow_forward2) Let M = {all postive integers), N = {0,1,2,3... 100), 0= {100,200,300,400,500). Determine if the following statements are true or false and explain your reasoning. a) NCM b) 0 C M c) O and N have at least one element in common d) O≤ N e) o≤o 1arrow_forward
- 4) Which of the following universal sets has W = {12,79, 44, 18) as a subset? Choose one. a) T = {12,9,76,333, 44, 99, 1000, 2} b) V = {44,76, 12, 99, 18,900,79,2} c) Y = {76,90, 800, 44, 99, 55, 22} d) x = {79,66,71, 4, 18, 22,99,2}arrow_forward3) What is the universal set that contains all possible integers from 1 to 8 inclusive? Choose one. a) A = {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8} b) B={-1,0,1,2,3,4,5,6,7,8} c) C={1,2,3,4,5,6,7,8} d) D = {0,1,2,3,4,5,6,7,8}arrow_forward5) 8.4 6.3 ?arrow_forward
- Wendy is looking over some data regarding the strength, measured in Pascals (Pa), of some rope and how the strength relates to the number of woven strands in the rope. The data are represented by the exponential function f(x) = 2x, where x is the number of woven strands. Explain how she can convert this equation to a logarithmic function when strength is 256 Pascals. Please type out answerarrow_forwardHarrison and Sherrie are making decisions about their bank accounts. Harrison wants to deposit $200 as a principal amount, with an interest of 2% compounded quarterly. Sherrie wants to deposit $200 as the principal amount, with an interest of 4% compounded monthly. Explain which method results in more money after 2 years. Show all work. Please type out answerarrow_forwardMike is working on solving the exponential equation 37x = 12; however, he is not quite sure where to start. Solve the equation and use complete sentences to describe the steps to solve. Hint: Use the change of base formula: log y = log y log barrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Orthogonality in Inner Product Spaces; Author: Study Force;https://www.youtube.com/watch?v=RzIx_rRo9m0;License: Standard YouTube License, CC-BY
Abstract Algebra: The definition of a Group; Author: Socratica;https://www.youtube.com/watch?v=QudbrUcVPxk;License: Standard Youtube License