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abstract algebra
11. Modular arithmetic a. Write the table for modular addition (modulo 7) b. Write...
a. b. c. What does the ciphertext ONL decode to with the modular inverse matrix from Question b? d. We use an encoded...
a.
b.
c. What does the ciphertext ONL decode to with the modular
inverse matrix from Question b?
d. We use an encoded text using a Caesar cipher. The ciphertext
was intercepted which is: THUBYLDH. What is the word? How did you
work this out?
Encode the uppercase letters of the English alphabet as A-0, B-1, C-2 and so on. Encrypt the word BUG with the block cipher matrix 16 4 11] 10 3 2 using modulo arithmetic with modulus...
Abstract Algebra I 1. Write down the Cayley table for the group generated -1 0 0...
Abstract Algebra I
1. Write down the Cayley table for the group generated -1 0 0 1 by the matrices 1 and 1. 2. Write down the Cayley table for the permutation group generated by the permutations (12)(34) and (13) in S 4 3. What do you notice about the two Cayley tables? How do they compare with the Cayley table for Z/8Z? How about the Cayley table for the square?
1. Write down the Cayley table for the group...
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R =...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table...
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table for G/H. Is G/H isomorphic to Z4 or Z2 x Z ? Justify your answer. 12. Show that G = {1, 7, 17, 23, 49, 55, 65, 71} is a group under multiplication modulo 96. Then express G as an external and an internal direct product of cyclic groups.
B3 a. Solve for x in this equation: 2x + 11 = 2 (mod 4). b....
B3 a. Solve for x in this equation: 2x + 11 = 2 (mod 4). b. What are the sets of units and zero divisors in the ring of integers modulo 22? (Specify at least the smaller set using set-roster notation.) c. Find a formula for the quotient and the exact remainder when 534 is divided by 8. Hint: find the remainder first by modular arithmetic. Then subtract the remainder from the power and divide to find the quotient.
g 2 is a primitive root modulo 19. Use the following table to assist you in...
g 2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t 1234567 8 9 10 11 12 13 14 15 16 1718 2 48 16 13 7 149 18 1715 11 3 6 12 5 10 1 (a) Find...
Explaining Expressions for Arithmetic arithmetic sequence whose first entry is 5 and s CA-208 Chapter 9...
Explaining Expressions for Arithmetic arithmetic sequence whose first entry is 5 and s CA-208 Chapter 9 Algebra Class Activity 9BB Sequences increases by 3 Entry Number Entry Ist 2nd 3rd 4th 5th Nth I1 14 17 a. If there were a Oth entry, what would it be? Put it in the table. b. Fill in the blanks to describe how to get entries in the sequence by starting from the Oth entr To find the Ist entry: Start at and...
Write a program to compute the product of two positive integers by repeated addition. Note: You...
Write a program to compute the product of two positive integers by repeated addition. Note: You are not allowed to use either multiplication or division operator. Understanding the Meaning of Multiplication: Multiplication of whole numbers can be thought of as repeated addition. // not allowed to use multiplication or division operator For example, suppose that a parking lot has 6 rows of parking spaces with 8 spaces in each row. How many parking spaces are in the lot? rows To...
G-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the f...
g-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cycl...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
a.
b.
c. What does the ciphertext ONL decode to with the modular
inverse matrix from Question b?
d. We use an encoded text using a Caesar cipher. The ciphertext
was intercepted which is: THUBYLDH. What is the word? How did you
work this out?
Encode the uppercase letters of the English alphabet as A-0, B-1, C-2 and so on. Encrypt the word BUG with the block cipher matrix 16 4 11] 10 3 2 using modulo arithmetic with modulus...
Abstract Algebra I
1. Write down the Cayley table for the group generated -1 0 0 1 by the matrices 1 and 1. 2. Write down the Cayley table for the permutation group generated by the permutations (12)(34) and (13) in S 4 3. What do you notice about the two Cayley tables? How do they compare with the Cayley table for Z/8Z? How about the Cayley table for the square?
1. Write down the Cayley table for the group...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table for G/H. Is G/H isomorphic to Z4 or Z2 x Z ? Justify your answer. 12. Show that G = {1, 7, 17, 23, 49, 55, 65, 71} is a group under multiplication modulo 96. Then express G as an external and an internal direct product of cyclic groups.
B3 a. Solve for x in this equation: 2x + 11 = 2 (mod 4). b. What are the sets of units and zero divisors in the ring of integers modulo 22? (Specify at least the smaller set using set-roster notation.) c. Find a formula for the quotient and the exact remainder when 534 is divided by 8. Hint: find the remainder first by modular arithmetic. Then subtract the remainder from the power and divide to find the quotient.
g 2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t 1234567 8 9 10 11 12 13 14 15 16 1718 2 48 16 13 7 149 18 1715 11 3 6 12 5 10 1 (a) Find...
Explaining Expressions for Arithmetic arithmetic sequence whose first entry is 5 and s CA-208 Chapter 9 Algebra Class Activity 9BB Sequences increases by 3 Entry Number Entry Ist 2nd 3rd 4th 5th Nth I1 14 17 a. If there were a Oth entry, what would it be? Put it in the table. b. Fill in the blanks to describe how to get entries in the sequence by starting from the Oth entr To find the Ist entry: Start at and...
g-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126...
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