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Math Algebra Elements Of Modern Algebra

If H and K are both normal subgroups of G , prove that H K is a normal subgroup of G .

If H and K are both normal subgroups of G , prove that H K is a normal subgroup of G .

Solution Summary: The author proves that HK is a normal subgroup of group G.

Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN: 9781285965918

Author: Gilbert

Publisher: Cengage

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Chapter 4.5, Problem 22E

If H and K are both normal subgroups of G , prove that H K is a normal subgroup of G .

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Chapter 4.5, Problem 21E

Chapter 4.5, Problem 23E

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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. 23E Ch. 4.1 - Exercises In Section 3.3, the centralizer of an...Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E 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 - Prob. 1TFE Ch. 4.3 - Prob. 2TFE Ch. 4.3 - Prob. 3TFE Ch. 4.3 - Prob. 4TFE Ch. 4.3 - True or False Label each of the following...Ch. 4.3 - Prob. 6TFE Ch. 4.3 - The alternating group A4 on 4 elements is the same...Ch. 4.3 - Prob. 1E Ch. 4.3 - Prob. 2E Ch. 4.3 - Prob. 3E Ch. 4.3 - Prob. 4E Ch. 4.3 - Prob. 5E Ch. 4.3 - Prob. 6E Ch. 4.3 - Prob. 7E Ch. 4.3 - Prob. 8E Ch. 4.3 - Prob. 9E Ch. 4.3 - Prob. 10E Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - Prob. 13E Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.3 - Prob. 16E Ch. 4.3 - Prob. 17E Ch. 4.3 - Prob. 18E Ch. 4.3 - Prob. 19E Ch. 4.3 - Prob. 20E Ch. 4.3 - Prob. 21E Ch. 4.3 - Prob. 22E Ch. 4.3 - Construct a multiplication table for the group G...Ch. 4.3 - Prob. 24E 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 - Prob. 29E 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 - Prob. 8E 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 - Prob. 15E 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 - Prob. 21E 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 - Prob. 2TFE 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 - Prob. 7TFE 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 - Prob. 17E 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 - Prob. 20E 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 - Prob. 25E Ch. 4.5 - Prob. 26E 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 - Prob. 30E Ch. 4.5 - Prob. 31E Ch. 4.5 - Prob. 32E Ch. 4.5 - Prob. 33E Ch. 4.5 - Prob. 34E Ch. 4.5 - Show that An has index 2 in Sn, and thereby...Ch. 4.5 - Prob. 36E Ch. 4.5 - Prob. 37E Ch. 4.5 - Let n be appositive integer, n1. Prove by...Ch. 4.5 - Prob. 39E 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 - Prob. 2TFE 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 - Prob. 4E Ch. 4.6 - Prob. 5E 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 - Prob. 13E 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 - Prob. 16E Ch. 4.6 - Prob. 17E Ch. 4.6 - 18. If is a subgroup of the group such that for...Ch. 4.6 - Prob. 19E Ch. 4.6 - Prob. 20E Ch. 4.6 - Prob. 21E Ch. 4.6 - Prob. 22E Ch. 4.6 - Prob. 23E 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 - Prob. 33E Ch. 4.6 - Prob. 34E Ch. 4.6 - Prob. 35E Ch. 4.6 - Prob. 36E Ch. 4.6 - Let H and K be arbitrary groups and let HK denotes...Ch. 4.6 - Prob. 38E Ch. 4.7 - True or False Label each of the following...Ch. 4.7 - Prob. 2TFE Ch. 4.7 - Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6...Ch. 4.7 - Prob. 2E Ch. 4.7 - Prob. 3E Ch. 4.7 - Prob. 4E Ch. 4.7 - Prob. 5E Ch. 4.7 - Prob. 6E Ch. 4.7 - Write 20 as the direct sum of two of its...Ch. 4.7 - Prob. 8E 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 - Prob. 12E 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 - Prob. 16E Ch. 4.7 - Prob. 17E Ch. 4.7 - Prob. 18E 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 - Prob. 21E Ch. 4.7 - Prob. 22E Ch. 4.7 - Prove that if r and s are relatively prime...Ch. 4.7 - Prob. 24E Ch. 4.8 - True or False Label each of the following...Ch. 4.8 - Prob. 2TFE Ch. 4.8 - Prob. 3TFE Ch. 4.8 - Prob. 4TFE Ch. 4.8 - Prob. 5TFE Ch. 4.8 - Prob. 6TFE Ch. 4.8 - Prob. 1E Ch. 4.8 - Prob. 2E 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 - Prob. 5E 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 - Prob. 8E Ch. 4.8 - 9. Determine which of the Sylow p-groups in each...Ch. 4.8 - Prob. 10E Ch. 4.8 - 11. Show that is a generating set for the...Ch. 4.8 - Prob. 12E 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 - Prob. 16E Ch. 4.8 - Prob. 17E Ch. 4.8 - Prob. 18E

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