Advanced Analysis

Abstract Algebra Portfolio - Draft

Due Friday, April 8 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 Cyclic groups.

1.1 Describe.

Explain the idea of a cyclic group in your own words. What is 〈a〉 and what does G = 〈a〉 mean? What does | a | mean? Give examples and non-examples, illustrations, comparisons, and observations. See Section 4.1. 1.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G

is cyclic and | G |= p2, where p is prime. (b) Prove that no group can have exactly two elements of order 2. (c) Let G be a cyclic group of order n and let H be the subgroup of order d. Show that

H = {x ∈ G | | x | divides d}. 1.3 Reflect.

Reflect on what you learned about cyclic groups. 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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2 Permutation Groups.

2.1 Describe.

Explain the idea of permutation groups in your own words. What is Sn, what is Dn, and howare they related? What is the definition of “permutation” and what are the different ways to represent a given permutation? Give examples and non-examples, illustrations, comparisons, and observations. See Section 5.1 and Section 5.2. 2.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Let G be a group of permutations on a set X. Let a ∈ X and define

stab(a) = {α ∈ G | α(a) = a}. Prove that stab(a) (called the stabilizer of a in G) is a subgroup of G.

(b) Let H = {β ∈ S5 | β(1) = 1 and β(3) = 3}. Prove that H is a subgroup of S5 and compute |H|.

(c) Prove that every element of Sn, n ≥ 2, can be written as a product of transpositions ofthe form (1k). (For example, σ = (1234) can be expressed as σ = (14)(13)(12).) 2.3 Reflect.

Reflect on what you learned about the mathematical concept of permutation groups. 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?

3 Cosets.

3.1 Describe.

What are cosets? What is special about the cosets of a subgroup? What does Lagrange’s Theorem say about cosets? Provide examples, comparisons to other mathematical ideas, and illustrations. See Section 6.1 and Section 6.2.

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3.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Let H and K be subgroups of a finite group G with H ⊆ K ⊆ G. Prove that

[G : H] = [G : K][K : H]. (b) Suppose that H and K are subgroups of a group G and there are elements a and b in G

such that aH ⊆ bK. Prove that H ⊆ K. (c) Let H be a subgroup of G and let a,b ∈ G. Show that aH = bH if and only if

Ha−1 = Hb−1. 3.3 Reflect.

Reflect on what you learned about the mathematical concept of cosets. 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?

4 Isomorphisms.

4.1 Describe.

What is an isomorphism? What does it mean to say two groups are isomorphic? Describe in your own words, provide examples of groups that are and are not isomorphic, provide examples of isomorphisms, comparisons to other mathematical ideas, and illustrations. See Section 3.3. 4.2 Prove.

Choose one of the following statements to prove. Your work must be your own! (a) Prove that S4 is not isomorphic to D12. (b) If G is a group, prove that Aut(G) and Inn(G) are groups. (c) Let ϕ : G → G′ be an isomorphism. Prove that if K is a subgroup of G, then

ϕ(K) = {ϕ(k) | k ∈ K} is a subgroup of G′.

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4.3 Reflect.

Reflect on what you learned about the mathematical concept of isomorphisms. 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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