## Abstract Algebra (Math 433), Fall 2004

### RSA and Cryptography

• RSA Calculator (use small messages to avoid overflow)
• RSA Security's Cryptographic Challenges

### Homework Exercises

Chapter Page Computational Theoretical
0 23 4, 16, 29, 38, 41, 46 8, 12, 14, 15, 21, 24
1 37 4, 13, 19, 22 6, 7, 8, 9
2 53 3, 5, 8, 13, 22, 24, 37 12, 14, 16, 17, 19, 29, 33
3 67 9, 24, 27, 28, 31, 33, 34, 41, 42 10, 13, 14, 16, 19, 21, 22
4 82 19, 22, 33, 40, 45, 46, 55, 65 24, 31, 41, 54, 56, 62, 64
5 111 4, 6, 24, 25, 32, 33, 36, 43 13, 16, 22, 31, 40, 45, 46
6 129 3, 4, 5, 7, 12, 16, 22, 23 2, 10, 30, 32, 33, 34, 35
7 145 1, 2, 3, 6, 8, 12, 13, 26, 33 10, 15, 19, 21, 23, 24, 36
8 162 5, 8, 12, 18, 20, 22, 26, 30, 40, 49, 51, 53 3, 14, 16, 55
9 186 3, 4, 5, 9, 12, 14, 21, 24 6, 10, 30, 37, 41, 43, 46, 48, 49, 58
10 205 2, 6, 10, 11, 14, 17, 19, 20 22, 37, 38, 42, 45, 46, 53 (see #7 p. 169)
11 219 3, 4, 6, 7, 8, 12, 15, 19, 25, 28 20, 30, 32, 34
24 407 5, 7, 11, 12, 18, 21, 44 1, 4, 10, 33, 36, 39, 42

1. Make your subject line exactly, exactly as follows: Math 433 Chapter X, where X is the chapter covered. Do not send your answers to my  beezer(at)ups(dot)edu address, but do send them to the address announced in class.
2. Put your full name as the first line of the body of your message.
3. Answer the questions in order, beginning each with the problem number.
4. Answers are due at 10:00 in the evening on the day before we begin discussing each new chapter (usually Monday night). They will not be accepted late.

[Chapter 0][Chapter 1][Chapter 2] [Chapter 3][Chapter 4][Chapter 5] [Chapter 6]

Chapter 0

1. Compute 17 mod 4.
2. Compute the greatest common divisor of 45 and 93.
3. The greatest common divisor of 19 and 85 is 1. Using this, find integers a and b so that 1 = 19a + 85b.
4. State carefully the three defining properties of an equivalence relation.
5. What is the big deal about equivalence relations? (Hint: partitions)

Chapter 1

1. What does the word "Abelian" mean? And why is it capitalized?
2. The group D4 is described carefully at the beginning of this chapter. What is the value of the product VH in D4?
3. Describe the identity element of D4.
4. How many elements are in the symmetry group of a hexagon?
5. Describe the symmetry group of a finite line segment.

Chapter 2

1. Name the three defining properties of a group.
2. What is the identity element of the group of complex n-th roots of unity?
3. In the group U(20), what is the inverse of 13?
4. Describe a typical element of SL(2,R).
5. Why was Heisenberg ashamed?

Chapter 3

1. What is the order of the element 3 in U(20)?
2. What is the order of the element 5 in U(23)?
3. Briefly compare and contrast the three subgroup tests.
4. List the elements of a non-trivial subgroup of U(11).
5. In words only, what is the center of a group?

Chapter 4

1. What is the Euler phi function?
2. Why is the notation | | used for both the order of a group and for the order of elements of a group?
3. Name the two cyclic groups.
4. What is a "subgroup lattice"?
5. What are the main conclusions of the Fundamental Theorem of Cyclic Groups?

Chapter 5

1. Express (1 3 4)(3 5 4) as a cycle, or a product of disjoint cycles.
2. What is a transposition?
3. What does it mean for a permutation to be even or odd?
4. Describe another group that is fundamentally the same as A3.
5. How many different ways are there to scramble Rubik's cube?

Chapter 6

1. Explain the Greek roots of the word "isomorphism."
2. Considering the set of complex numbers as a plane, what is the surprising relationship between the unit circle and the punctured plane?
3. List three properties of group elements that are preserved by isomorphisms.
4. List three properties of groups that are preserved by isomorphisms.
5. The group Aut(Zn) is isomorphic to which familiar group?

Chapter 7

1. What is a coset?
2. State Lagrange's Theorem, in your own words.
3. How many groups are there of order 23? Why?
4. How many groups are there of order 46? Why?
5. Describe the group of symmetries of a cube.

Chapter 8
I'll use the symbol "(+)" as a replacement for the direct product symbol.

1. Suppose that G and H are both cyclic groups. When is G (+) H cyclic?
2. Write down the two groups of order 4, making use of the construction in this chapter.
3. What is the order of (3,6) as an element of Z5 (+) Z12?
4. Write U(21) as a direct product of two U( ) groups.
5. Now write U(21) as a direct product of two Z groups.

Chapter 9

1. What is so interesting about normal subgroups?
2. What is a factor group?
3. 8Z is a subgroup of Z. In Z/8Z compute (3+8Z)+(7+8Z).
4. What can be said about G when G/Z(G) is cyclic?
5. What is the difference between an external direct product and an internal direct product?

Chapter 10

1. What is the kernel of a homomorphism?
2. What is the relationship between kernels and normal subgroups?
3. State three element properties of group homomorphisms.
4. State three subgroup properties of group homomorphisms.
5. What is the First Isomorphism Theorem?

Chapter 11

1. How many abelian groups are there of order 200 = 23 52?
2. How many abelian groups are there of order 729=36?
3. An abelian group of order 72 contains an element of order 8. What are the possibilities for this group?
4. The group Z8 (+) Z3 (+) Z3 must contain a subgroup of order 6. Find this subgroup.
5. When was the classification of the finite abelian groups first given, and who did it?

Chapter 24

1. State Sylow's First Theorem.
2. How many groups are there of order 49? Why?
3. How many groups are there of order 69? Why?
4. What's all the fuss about Sylow's Theorems?
5. Name one of Sylow's academic great-great-great-great-great-great-grandchildren. (That's (great)6grand-children.)

This is: http://buzzard.ups.edu/courses/2004fall/m433f2004.html
Maintained by: Rob Beezer
Last updated: August 23, 2004