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to about 56 million cosets for testing. Cosets played a role in this effort because Rokicki's program could handle the 19.5+ billion elements in 156 the same coset in about 20 seconds. Exercises HOMER SIMPSON I don't know, Marge. Trying is the first step towards failure. 1. Let H = {(1), (12)(34), (13)(24), (14)(23) }. Find the left cosets of Hin A, (see Table 5.1 on page 111). 2. Let H be as in Exercise 1. How many left cosets of H in S a there? (Determine this without listing them.) 3. Let H = {0, +3, +6, +9, . .. }. Find all the left cosets of H in Z 4. Rewrite the condition a b E H given in property 5 of the lemma on page 145 in additive notation. Assume that the group is Abelian. are 5. Let H be as in Exercise 3. Use Exercise 4 to decide whether or not the following cosets of H are the same. a. 11 + Hand 17 + H b. -1+ H and 5 + H c. 7+Hand 23 + H 6. Let n be a positive integer. Let H = {0, +n , + 2n , +3n, . . .}. Find all left cosets of H in Z. How many are there? 7. Find all of the left cosets of {1, 11} in U(30). 8 Suppose that a has order 15. Find all of the left cosets of (a) in (a). 9. Let lal 30. How many left cosets of (at) in (a) are there? List them. 10. Give an example of a group G and subgroups H and K such that HK = {hE H, k E K) is not a subgroup of G. 11. If H and K are subgroups of G and g belongs to G, show that g(Hn K) gH n gK. 12. Let a and b be nonidentity elements of different orders in a group G of order 155, Prove that the only subgroup of G that contains a and b is G itself. 13. Let H be a subgroup of R, the group of nonzero real numbers un- der multiplication. If R*C HCR, prove that H = R* or H R 14. Let C be the group of nonzero complex numbers under multiplica- tion and let H= ta+ bi E CI a +b= 1). Give a geometric de- scription of the coset (3+ 4i)H. Give a geometric description of the र 3 3 coset (c + di)H. 32

to about 56 million cosets for testing. Cosets played a role in this effort because Rokicki's program could handle the 19.5+ billion elements in 156 the same coset in about 20 seconds. Exercises HOMER SIMPSON I don't know, Marge. Trying is the first step towards failure. 1. Let H = {(1), (12)(34), (13)(24), (14)(23) }. Find the left cosets of Hin A, (see Table 5.1 on page 111). 2. Let H be as in Exercise 1. How many left cosets of H in S a there? (Determine this without listing them.) 3. Let H = {0, +3, +6, +9, . .. }. Find all the left cosets of H in Z 4. Rewrite the condition a b E H given in property 5 of the lemma on page 145 in additive notation. Assume that the group is Abelian. are 5. Let H be as in Exercise 3. Use Exercise 4 to decide whether or not the following cosets of H are the same. a. 11 + Hand 17 + H b. -1+ H and 5 + H c. 7+Hand 23 + H 6. Let n be a positive integer. Let H = {0, +n , + 2n , +3n, . . .}. Find all left cosets of H in Z. How many are there? 7. Find all of the left cosets of {1, 11} in U(30). 8 Suppose that a has order 15. Find all of the left cosets of (a) in (a). 9. Let lal 30. How many left cosets of (at) in (a) are there? List them. 10. Give an example of a group G and subgroups H and K such that HK = {hE H, k E K) is not a subgroup of G. 11. If H and K are subgroups of G and g belongs to G, show that g(Hn K) gH n gK. 12. Let a and b be nonidentity elements of different orders in a group G of order 155, Prove that the only subgroup of G that contains a and b is G itself. 13. Let H be a subgroup of R, the group of nonzero real numbers un- der multiplication. If R*C HCR, prove that H = R* or H R 14. Let C be the group of nonzero complex numbers under multiplica- tion and let H= ta+ bi E CI a +b= 1). Give a geometric de- scription of the coset (3+ 4i)H. Give a geometric description of the र 3 3 coset (c + di)H. 32

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.2: Exponential Functions
Problem 54E
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to about 56 million cosets for testing. Cosets played a role in this effort because Rokicki's program could handle the 19.5+ billion elements in 156 the same coset in about 20 seconds. Exercises HOMER SIMPSON I don't know, Marge. Trying is the first step towards failure. 1. Let H = {(1), (12)(34), (13)(24), (14)(23) }. Find the left cosets of Hin A, (see Table 5.1 on page 111). 2. Let H be as in Exercise 1. How many left cosets of H in S a there? (Determine this without listing them.) 3. Let H = {0, +3, +6, +9, . .. }. Find all the left cosets of H in Z 4. Rewrite the condition a b E H given in property 5 of the lemma on page 145 in additive notation. Assume that the group is Abelian. are 5. Let H be as in Exercise 3. Use Exercise 4 to decide whether or not the following cosets of H are the same. a. 11 + Hand 17 + H b. -1+ H and 5 + H c. 7+Hand 23 + H 6. Let n be a positive integer. Let H = {0, +n , + 2n , +3n, . . .}. Find all left cosets of H in Z. How many are there? 7. Find all of the left cosets of {1, 11} in U(30). 8 Suppose that a has order 15. Find all of the left cosets of (a) in (a). 9. Let lal 30. How many left cosets of (at) in (a) are there? List them. 10. Give an example of a group G and subgroups H and K such that HK = {hE H, k E K) is not a subgroup of G. 11. If H and K are subgroups of G and g belongs to G, show that g(Hn K) gH n gK. 12. Let a and b be nonidentity elements of different orders in a group G of order 155, Prove that the only subgroup of G that contains a and b is G itself. 13. Let H be a subgroup of R, the group of nonzero real numbers un- der multiplication. If R*C HCR, prove that H = R* or H R 14. Let C be the group of nonzero complex numbers under multiplica- tion and let H= ta+ bi E CI a +b= 1). Give a geometric de- scription of the coset (3+ 4i)H. Give a geometric description of the र 3 3 coset (c + di)H. 32
Transcribed Image Text:to about 56 million cosets for testing. Cosets played a role in this effort because Rokicki's program could handle the 19.5+ billion elements in 156 the same coset in about 20 seconds. Exercises HOMER SIMPSON I don't know, Marge. Trying is the first step towards failure. 1. Let H = {(1), (12)(34), (13)(24), (14)(23) }. Find the left cosets of Hin A, (see Table 5.1 on page 111). 2. Let H be as in Exercise 1. How many left cosets of H in S a there? (Determine this without listing them.) 3. Let H = {0, +3, +6, +9, . .. }. Find all the left cosets of H in Z 4. Rewrite the condition a b E H given in property 5 of the lemma on page 145 in additive notation. Assume that the group is Abelian. are 5. Let H be as in Exercise 3. Use Exercise 4 to decide whether or not the following cosets of H are the same. a. 11 + Hand 17 + H b. -1+ H and 5 + H c. 7+Hand 23 + H 6. Let n be a positive integer. Let H = {0, +n , + 2n , +3n, . . .}. Find all left cosets of H in Z. How many are there? 7. Find all of the left cosets of {1, 11} in U(30). 8 Suppose that a has order 15. Find all of the left cosets of (a) in (a). 9. Let lal 30. How many left cosets of (at) in (a) are there? List them. 10. Give an example of a group G and subgroups H and K such that HK = {hE H, k E K) is not a subgroup of G. 11. If H and K are subgroups of G and g belongs to G, show that g(Hn K) gH n gK. 12. Let a and b be nonidentity elements of different orders in a group G of order 155, Prove that the only subgroup of G that contains a and b is G itself. 13. Let H be a subgroup of R, the group of nonzero real numbers un- der multiplication. If R*C HCR, prove that H = R* or H R 14. Let C be the group of nonzero complex numbers under multiplica- tion and let H= ta+ bi E CI a +b= 1). Give a geometric de- scription of the coset (3+ 4i)H. Give a geometric description of the र 3 3 coset (c + di)H. 32
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