Advanced Analysis

Abstract Algebra Portfolio - Draft 1

Due Friday, March 2

These tasks are intended to both develop and demonstrate the course objectives. They are designed to push you – think of them as really hard workouts for your brain. Please ask for help when you need it! This is also an opportunity for you to make connections across the semester and practice your proof writing.

You should turn in a typed (or very, very neatly written!) and organized draft on Gradescope. Please label your problem selections clearly!

1 Modular arithmetic.

1.1 Describe.

Explain the idea of modular arithmetic in your own words. What does a ≡ b mod n mean? Why is this relation an equivalence relation? What are the associated equivalence classes? Give examples, illustrations, and comparisons. See Example 1.30 and The Integers mod n, including Example 3.1, Example 3.2, and Proposition 3.4. 1.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Let n ∈ N. Use the division algorithm to prove that every integers is congruent mod n to

precisely one of the integers 0, 1, . . . , n−1. Conclude that if r is an integer, then there is exactly one s ∈ Z such that 0 ≤ s < n and [r] = [s]. Hence, the integers are indeed partitioned by congruence mod n.

(b) Show that addition and multiplication mod n are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod n.

(c) Multiplication distributes over addition modulo n: a(b + c) ≡ ab + ac mod n.

1.3 Reflect.

Reflect on what you learned about modular arithmetic. Some questions or ideas you might consider include:

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• What aspects of this topic are you curious to know more about? Give one or more examples of questions about the material that you’d like to explore further, and describe why these are interesting questions to you.

• Take one problem you have worked on related to this topic that you struggled to understand and solve, and explain how the struggle itself was valuable to your learning.

• How did this topic enlarge your sense of what it means to do mathematics? • Describe an instance where you struggled with this topic, and initially had the wrong idea, but then later realized your error. In this instance, in what ways was a struggle or mistake valuable to your eventual understanding?

2 Group fundamentals.

2.1 Describe.

Explain the idea of a group in your own words. What is the definition? Give examples and non-examples, illustrations, comparisons, and observations. See Section 3.2. 2.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Let G be a finite group with identity e. Then the number of elements x of G such that

x3 = e is odd, and the number of elements x of G such that x2 6= e is even. (b) If a group G satisfies the following: if a, b, c ∈ G and ab = ca, then b = c. Then G is abelian. (c) A group is abelian if and only if (ab)−1 = a−1b−1 for all a and b in the group.

2.3 Reflect.

Reflect on what you learned about the mathematical concept of a group. Some questions or ideas you might consider include:

• What aspects of this topic are you curious to know more about? Give one or more examples of questions about the material that you’d like to explore further, and describe why these are interesting questions to you.

• Take one problem you have worked on related to this topic that you struggled to understand and solve, and explain how the struggle itself was valuable to your learning.

• How did this topic enlarge your sense of what it means to do mathematics? • Describe an instance where you struggled with this topic, and initially had the wrong idea, but then later realized your error. In this instance, in what ways was a struggle or mistake valuable to your eventual understanding?

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3 Subgroups.

3.1 Describe.

What is a subgroup? How can a subgroup be identified? Provide examples and non-examples, comparisons to other mathematical ideas, and illustrations. See Section 3.3. 3.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Let R× be the group of nonzero real numbers under multiplication. Then

H = {x ∈ R× | x2 is rational } is a subgroup of R×. (b) Prove or disprove: If H and K are subgroups of a group G, then

HK = {hk : h ∈ Handk ∈ K} is a subgroup of G. What if G is abelian? (c) Let H be a subgroup of G and C(H) = {g ∈ G : gh = hg∀h ∈ H}. Prove that C(H) is a

subgroup of G. (This subgroup is called the centralizer of H in G.) 3.3 Reflect.

Reflect on what you learned about the mathematical concept of a subgroup. Some questions or ideas you might consider include:

• What aspects of this topic are you curious to know more about? Give one or more examples of questions about the material that you’d like to explore further, and describe why these are interesting questions to you.

• Take one problem you have worked on related to this topic that you struggled to understand and solve, and explain how the struggle itself was valuable to your learning.

• How did this topic enlarge your sense of what it means to do mathematics? • Describe an instance where you struggled with this topic, and initially had the wrong idea, but then later realized your error. In this instance, in what ways was a struggle or mistake valuable to your eventual understanding?

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