Reading Questions
Associated with Judson's Abstract Algebra
Chapter 1, Preliminaries
- What do relations and mappings have in common?
- What makes relations and mappings different?
- Give the definitions of the three defining properties of an equivalence
relation.
- What is the big deal about equivalence relations? (Hint: Partitions.)
- Describe a general technique for proving that two sets are equal.
Chapter 2, The Integers
- Use Sage to express 123456792 as a product of primes.
- Find the greatest common divisor of 84 and 52.
- Find integers \(r\) and \(s\) so that \(r(84)+s(52)= \gcd(84,52) \).
- Explain the use of the term "induction hypothesis".
- What is Goldbach's Conjecture? And why is it called a conjecture?
Chapter 3, Groups
- In the group \(\mathbb{Z}_{8}\), compute (a) \(6+7, \quad \) (b) \(2^{-1}
\)
- In the group \(U(16)\), compute (a) \(5 \cdot 7, \quad \) (b) \(3^{-1} \)
- State the definition of a group.
- Explain a single method that will decide if a subset of a group is itself
a subgroup.
- Explain the origin of the term "abelian'' for a commutative group.
Chapter 4, Cyclic Groups
- What is the order of the element \(3 \) in \(U(20) \)?
- What is the order of the element \(5 \) in \(U(23) \)?
- Find three generators of \(\mathbb{Z}_{8}\).
- Find three generators of the \(5^{\text{th}}\) roots of unity.
- Show how to compute \(15^{40} (\mod 23) \) efficiently by hand. Check
your answer with Sage.
Chapter 5, Permutation Groups
- 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.)
- What is a transposition?
- What does it mean for a permutation to be even or odd?
- Describe another group that is fundamentally the same as \(A_3 \).
- Write the elements of the symmetry group of a pentagon using permutations
in cycle notation.
Chapter 6, Cosets and Lagrange's Theorem
- State Lagrange's Theorem in your own words.
- Determine the left cosets of \( \langle 3 \rangle\) in
\(\mathbb{Z}_{9}\).
- 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 \(S_4 \)?
- Suppose \(G\) is a group of order \(29\). Describe \(G\).
- \(p=137909\) is a prime. Explain how to compute \(57^{137909} (\mod
137909) \) without a calculator.
Chapter 7, Introduction to Cryptography
- Use the
euler_phi()
function in Sage to compute \(\phi(893\
456\ 123) \).
- Use the
power_mod()
function in Sage to compute \( 7^{324}
(\mod{895}) \).
- 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.
- Explain how, in RSA, message encoding differs from message
verification.
- Explain how one could be justified in saying that Diffie and Hellman's
proposal in 1976 was "revolutionary".
Chapter 9, Isomorphisms
- Determine the order of \( (1, 2) \) in \(\mathbb{Z}_{4}
\times\mathbb{Z}_{8} \).
- List three properties of a group that are preserved by an
isomorphism.
- find a group isomorphic to \(\mathbb{Z}_{15} \) that is an external
direct product of two non-trivial groups.
- Explain why we can now say "the infinite cyclic group".
- Compare and contrast external direct products and internal direct
products.
Chapter 10, Normal Subgroups and Factor Groups
- 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 \( G \).
- Based on your answer to the previous question, is \( H \) normal in \( G
\)? Explain why or why not.
- \(\mathbb{Z}_{8} \) is a normal subgroup in \( \mathbb{Z} \) . In the
factor group \( \mathbb{Z}/ \mathbb{Z}_{8} \), perform the computation
\((3+8\mathbb{Z})+(7+8\mathbb{Z}) \).
- List two statements about a group \(G\) and a subgroup \( H \) that are
equivalent to "\(H\) is normal in \(G\)."
- In your own words, what is a factor group?
Chapter 11, Homomorphisms
- Consider the function \(\phi:\mathbb{Z}_{10} \rightarrow \mathbb{Z}_{10}
\) defined by \(\phi(x) = x+x \). Prove that \(\phi \) is a group
homomorphism.
- For \(\phi \) definied in the previous question, explain why \( \phi\) is
not a group isomorphism.
- Compare and contrast isomorphisms and homomorphisms.
- Paraphrase the First Isomorphism Theorem using only words. No
symbols allowed at all.
- "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
- How many abelian groups are there of order \(200=2^3 5^2 \)?
- How many abelian groups are there of order \(729 = 3^6 \)?
- Find a subgroup of order \( 6\) in \(\mathbb{Z}_{8} \times \mathbb{Z}_{3}
\times \mathbb{Z}_{3}\).
- It can be shown that an abelian group of order \(72\) contains a subgroup
of order \(8\). What are the possibilities for this subgroup?
- What is a principal series of the group \(G\)?
Chapter 14, Group Actions
- Give an informal description of a group action.
- Describe the class equation.
- What are the groups of order \(49\)?
- How many switching functions are there with \(5\) inputs?
- The "Historical Note" mentions the proof of Burnside's Conjecture. How
long was the proof?
Chapter 15, The Sylow Theorems
- State Sylow's First Theorem.
- How many groups are there of order \(69\)? Why?
- Give two descriptions, different in character, of the normalizer of a
subgroup.
- What's all the fuss about Sylow's Theorems?
- Name one of Sylow's academic
great-great-great-great-great-great-grandchildren. (That's
\(\text{(great-)}^6\text{grandchildren.)}\)