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8th Edition

Gilbert + 2 others

Publisher: Cengage Learning,

ISBN: 9781285463230

Chapter 4.3, Problem 7TFE

Textbook Problem

The alternating group

Expert Solution

To determine

Whether the statement, “The alternating group

Consider the statement, “The alternating group

The order of alternating group

Elements Of Modern Algebra

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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 - Exercises Prove that in any group, the relation x...Ch. 4.1 - Exercises In Section 3.3, the centralizer of an...Ch. 4.1 - Exercises
25. Let be a permutation of a set . For...Ch. 4.1 - Exercises
26. Consider the symmetric group. Find...Ch. 4.1 - Exercises
27. Consider the alternating group. Find...Ch. 4.1 - Exercises
28. Consider the octic group . Find each...Ch. 4.1 - Exercises
29. A subgroup of the group is called...Ch. 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 - True or False Label each of the following...Ch. 4.3 - True or False Label each of the following...Ch. 4.3 - True or False Label each of the following...Ch. 4.3 - True or False Label each of the following...Ch. 4.3 - True or False
Label each of the following...Ch. 4.3 - The symmetric group on elements is the same as...Ch. 4.3 - The alternating group A4 on 4 elements is the same...Ch. 4.3 - Exercises List all elements in the group of...Ch. 4.3 - Exercises
List all elements in the group of...Ch. 4.3 - Exercises List all elements in the group of...Ch. 4.3 - Exercises List all elements in the group of...Ch. 4.3 - Exercises
Determine wether the given figure has...Ch. 4.3 - Exercises Determine wether the given figure has...Ch. 4.3 - Exercises Determine wether the given figure has...Ch. 4.3 - Exercises Determine wether the given figure has...Ch. 4.3 - Exercises
Determine wether the given figure has...Ch. 4.3 - Exercises Determine wether the given figure has...Ch. 4.3 - Exercises
Describe the elements in the group of...Ch. 4.3 - Exercises
Describe the elements in the group of...Ch. 4.3 - Exercises Describe the elements in the group of...Ch. 4.3 - Exercises Describe the elements in the group of...Ch. 4.3 - Exercises Describe the elements in the group of...Ch. 4.3 - Exercises
Describe the elements in the group of...Ch. 4.3 - Describe the elements in the groups of symmetries...Ch. 4.3 - Describe the elements in the groups of symmetries...Ch. 4.3 - Describe the elements in the groups of symmetries...Ch. 4.3 - Describe the elements in the groups of symmetries...Ch. 4.3 - Show that the group of symmetries in Example 3 of...Ch. 4.3 - Construct a multiplication table for the group of...Ch. 4.3 - Construct a multiplication table for the group G...Ch. 4.3 - Construct a multiplication table for the group G...Ch. 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 - Find an isomorphism from the group in Exercise 23...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 - 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 -
Ch. 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 - For an arbitrary subgroup of the group , define...Ch. 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 - Find all subgroups of the alternating group .
Ch. 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 -
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 - True or False
Label each of the following...Ch. 4.5 - True or False Label each of the following...Ch. 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 - If H,L, is a collection of normal subgroups H of...Ch. 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 -
20.With and as in Exercise 18, prove that is a...Ch. 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 - 25. Find the center of the octic group .
Ch. 4.5 - Find the center of A4.Ch. 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 - 30. Find the normalizer of the subgroup of the...Ch. 4.5 - Let H be a subgroup of G. Define the relation...Ch. 4.5 - Describe the equivalence classes in Exercise 31....Ch. 4.5 - Let n1 in the group of integers under addition,...Ch. 4.5 - 34. Let be a subgroup of with index .
a....Ch. 4.5 - Show that An has index 2 in Sn, and thereby...Ch. 4.5 - 36. Let be a nonempty subset of a group . Prove...Ch. 4.5 - Find the subgroup of Sn that is generated by the...Ch. 4.5 - Let n be appositive integer, n1. Prove by...Ch. 4.5 - 39. Let be a group and . Prove that , the set...Ch. 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 - 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 - 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 - In Exercises 16, H is a normal subgroup of the...Ch. 4.6 - In Exercises 16, H is a normal subgroup of the...Ch. 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 - Let G=S3. For each H that follows, show that the...Ch. 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 - 16. Repeat Exercise with the quaternion group ,...Ch. 4.6 - 17. Repeat Exercise where is the multiplicative...Ch. 4.6 - 18. If is a subgroup of the group such that for...Ch. 4.6 - 19. Let be a subgroup of the group . Prove that ...Ch. 4.6 - If H is a normal subgroup of the group G, prove...Ch. 4.6 - Let H be a normal subgroup of finite group G. If...Ch. 4.6 - Let H be a normal subgroup of the group G. Prove...Ch. 4.6 - 23. Let be a torsion group, as defined in...Ch. 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 - 33. (See Exercise 32.) Let be a group with center...Ch. 4.6 - If H and K are normal subgroups of the group G...Ch. 4.6 - 35. (See Exercise 34.) If , prove that each...Ch. 4.6 - Let H be a subgroup of G and let K be a normal...Ch. 4.6 - Let H and K be arbitrary groups and let HK denotes...Ch. 4.6 -
38. (See Exercise 37.) Let and be fixed elements...Ch. 4.7 - True or False Label each of the following...Ch. 4.7 - True or False Label each of the following...Ch. 4.7 - Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6...Ch. 4.7 - 2. Let and be subgroups of the abelian group ...Ch. 4.7 - 3. Let and be subgroups of the abelian group ...Ch. 4.7 - Let H1={ [ 0 ],[ 7 ],[ 14 ] } and H2={ [ 0 ],[ 3...Ch. 4.7 - Let H1={ [ 0 ],[ 15 ] },H2={ [ 0 ],[ 10 ],[ 20 ]...Ch. 4.7 - 6. Let and be subgroups of the abelian group ...Ch. 4.7 - Write 20 as the direct sum of two of its...Ch. 4.7 - 8. Write as the direct sum of two of its...Ch. 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 - 12. Assume that are subgroups of the abelian...Ch. 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 - 16. (This is additive version of Exercise 37 in...Ch. 4.7 - (See Exercise 16.) Find the order of each of the...Ch. 4.7 - 18. a. Find all subgroups of .
b. Find all...Ch. 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 - Suppose a is an element of order rs in an abelian...Ch. 4.7 - (See Exercise21.) Assume that a is an element of...Ch. 4.7 - Prove that if r and s are relatively prime...Ch. 4.7 - Prove Theorem 4.35: If H1,H2,...,Hn are subgroups...Ch. 4.8 - True or False Label each of the following...Ch. 4.8 - True or False
Label each of the following...Ch. 4.8 - True or False
Label each of the following...Ch. 4.8 - True or False
Label each of the following...Ch. 4.8 - True or False Label each of the following...Ch. 4.8 - True or False Label each of the following...Ch. 4.8 - Give an example of a p-group of order .
Ch. 4.8 - 2. Find two p-groups of order that are not...Ch. 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 - 5. For each of the following , letbe the additive...Ch. 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 - 8. Let be a finite group, prime, and a Sylow...Ch. 4.8 - 9. Determine which of the Sylow p-groups in each...Ch. 4.8 - 10. Determine which of the Sylow 3-grops in...Ch. 4.8 - 11. Show that is a generating set for the...Ch. 4.8 - 12. Give an example where is a finite nonabelian...Ch. 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 - 16. Suppose that is a cyclic group of order ,...Ch. 4.8 - 17. Prove that result in Exercise 16 for an...Ch. 4.8 - 18. Suppose that if is an abelian group of order...

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