Reading Questions

Associated with Judson's Abstract Algebra

Chapter 1, Preliminaries

  1. What do relations and mappings have in common?
  2. What makes relations and mappings different?
  3. Give the definitions of the three defining properties of an equivalence relation.
  4. What is the big deal about equivalence relations? (Hint: Partitions.)
  5. Describe a general technique for proving that two sets are equal.

Chapter 2, The Integers

  1. Use Sage to express 123456792 as a product of primes.
  2. Find the greatest common divisor of 84 and 52.
  3. Find integers \(r\) and \(s\) so that \(r(84)+s(52)= \gcd(84,52) \).
  4. Explain the use of the term "induction hypothesis".
  5. What is Goldbach's Conjecture? And why is it called a conjecture?

Chapter 3, Groups

  1. In the group \(\mathbb{Z}_{8}\), compute (a) \(6+7, \quad \) (b) \(2^{-1} \)
  2. In the group \(U(16)\), compute (a) \(5 \cdot 7, \quad \) (b) \(3^{-1} \)
  3. State the definition of a group.
  4. Explain a single method that will decide if a subset of a group is itself a subgroup.
  5. Explain the origin of the term "abelian'' for a commutative group.

Chapter 4, Cyclic Groups

  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. Find three generators of \(\mathbb{Z}_{8}\).
  4. Find three generators of the \(5^{\text{th}}\) roots of unity.
  5. Show how to compute \(15^{40} (\mod 23) \) efficiently by hand. Check your answer with Sage.

Chapter 5, Permutation Groups

  1. Express \( (1\ 3\ 4) (3\ 5\ 4)\) as a cycle, or a product of disjoint cycles. (Inerpret the composition of functions in the order used in Sage, which is the reverse of the order used in the book.)
  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 \(A_3 \).
  5. Write the elements of the symmetry group of a pentagon using permutations in cycle notation.

Chapter 6, Cosets and Lagrange's Theorem

  1. State Lagrange's Theorem in your own words.
  2. Determine the left cosets of \( \langle 3 \rangle\) in \(\mathbb{Z}_{9}\).
  3. The set \(\{(), (1\ 2)(3\ 4),(1\ 3)(2\ 4), (1\ 4)(2\ 3) \} \) is a subgroup of \(S_4 \). What is it's index in \(S_4 \)?
  4. Suppose \(G\) is a group of order \(29\). Describe \(G\).
  5. \(p=137909\) is a prime. Explain how to compute \(57^{137909} (\mod 137909) \) without a calculator.

Chapter 7, Introduction to Cryptography

  1. Use the euler_phi() function in Sage to compute \(\phi(893\ 456\ 123) \).
  2. Use the power_mod() function in Sage to compute \( 7^{324} (\mod{895}) \).
  3. Explain the mathematical basis for saying: encrypting a message using an RSA public key is very simple computationally, while decrypting a communciation without the private key is very hard computationally.
  4. Explain how, in RSA, message encoding differs from message verification.
  5. Explain how one could be justified in saying that Diffie and Hellman's proposal in 1976 was "revolutionary".

Chapter 9, Isomorphisms

  1. Determine the order of \( (1, 2) \) in \(\mathbb{Z}_{4} \times\mathbb{Z}_{8} \).
  2. List three properties of a group that are preserved by an isomorphism.
  3. find a group isomorphic to \(\mathbb{Z}_{15} \) that is an external direct product of two non-trivial groups.
  4. Explain why we can now say "the infinite cyclic group".
  5. Compare and contrast external direct products and internal direct products.

Chapter 10, Normal Subgroups and Factor Groups

  1. Let \(G\) be the group of symmetries of an equilateral triangle, expressed as permutations of the vertices numbered \(1,\ 2,\ 3 \). Let \( H\) be the subgroup \( H= \langle \{ (1\ 2) \} \rangle\). Build the left and right cosets of \( H \) in \( G \).
  2. Based on your answer to the previous question, is \( H \) normal in \( G \)? Explain why or why not.
  3. \(\mathbb{Z}_{8} \) is a normal subgroup in \( \mathbb{Z} \) . In the factor group \( \mathbb{Z}/ \mathbb{Z}_{8} \), perform the computation \((3+8\mathbb{Z})+(7+8\mathbb{Z}) \).
  4. List two statements about a group \(G\) and a subgroup \( H \) that are equivalent to "\(H\) is normal in \(G\)."
  5. In your own words, what is a factor group?

Chapter 11, Homomorphisms

  1. Consider the function \(\phi:\mathbb{Z}_{10} \rightarrow \mathbb{Z}_{10} \) defined by \(\phi(x) = x+x \). Prove that \(\phi \) is a group homomorphism.
  2. For \(\phi \) definied in the previous question, explain why \( \phi\) is not a group isomorphism.
  3. Compare and contrast isomorphisms and homomorphisms.
  4. Paraphrase the First Isomorphism Theorem using only words. No symbols allowed at all.
  5. "For every normal subgroup there is a homomorphism, and for every homomorphism there is a normal subgroup." Explain the (precise) basis for this (vague) statement.

Chapter 12, Matrix Groups and Symmetry

Covered only if time permits

Chapter 13, The Structure of Groups

  1. How many abelian groups are there of order \(200=2^3 5^2 \)?
  2. How many abelian groups are there of order \(729 = 3^6 \)?
  3. Find a subgroup of order \( 6\) in \(\mathbb{Z}_{8} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3}\).
  4. It can be shown that an abelian group of order \(72\) contains a subgroup of order \(8\). What are the possibilities for this subgroup?
  5. What is a principal series of the group \(G\)?

Chapter 14, Group Actions

  1. Give an informal description of a group action.
  2. Describe the class equation.
  3. What are the groups of order \(49\)?
  4. How many switching functions are there with \(5\) inputs?
  5. The "Historical Note" mentions the proof of Burnside's Conjecture. How long was the proof?

Chapter 15, The Sylow Theorems

  1. State Sylow's First Theorem.
  2. How many groups are there of order \(69\)? Why?
  3. Give two descriptions, different in character, of the normalizer of a subgroup.
  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 \(\text{(great-)}^6\text{grandchildren.)}\)