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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 4.8, Problem 10E
Textbook Problem
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Determine which of the Sylow 3-grops in Exercise 4 are normal.

Exercise 4.

Find all Sylow 3-subgroups of the symmetric group S 4 .

To determine

Which of the Sylow 3 groups of the symmetric group S4 are normal.

Explanation of Solution

Given information:

S4 is a symmetric group.

Formula used:

Let G be a finite group, p is prime, and H a Sylow p group, H is normal in G if and only if H is the only H is the only Sylow p group in G.

Calculation:

Let a group be G=S4.

S4={(1),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(1,2,3),(1,2,4),(1,3,4),(2,3,4),(1,3,2),(1,4,2),(1,4,3),(2,4,3),(1,2,3,4),(1,3,4,2),(1,4,2,3),(4,3,2,1),(2,4,3,1),(3,2,4,1),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)}

|G|=24=233

Here 3 divides |G| but 32=9 does not divide |G|

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Chapter 4 Solutions

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 - 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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