Advanced Analysis

need the question highlight done 

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:

1 of ??

http://abstract.ups.edu/aata/section-sets-and-equivalence-relations.html#Hga

http://abstract.ups.edu/aata/section-mod-n-sym.html#FiW

http://abstract.ups.edu/aata/section-mod-n-sym.html#rPl

http://abstract.ups.edu/aata/section-mod-n-sym.html#XWu

http://abstract.ups.edu/aata/section-mod-n-sym.html#LIc

Carlos Rivera

Carlos Rivera

• 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?

2 of ??

http://abstract.ups.edu/aata/section-groups-define.html#TmB

Carlos Rivera

Carlos Rivera

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?

3 of ??

http://abstract.ups.edu/aata/section-subgroups.html#dLG

Carlos Rivera

Carlos Rivera

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?

1 of 4

http://abstract.ups.edu/aata/section-cyclic-subgroups.html

Carlos Rivera

Carlos Rivera

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.

2 of 4

http://abstract.ups.edu/aata/section-permutation-definitions.html

http://abstract.ups.edu/aata/section-dihedral-groups.html

http://abstract.ups.edu/aata/section-cosets.html

http://abstract.ups.edu/aata/section-lagranges-theorem.html

Carlos Rivera

Carlos Rivera

Carlos Rivera

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′.

3 of 4

http://abstract.ups.edu/aata/section-subgroups.html#dLG

Carlos Rivera

Carlos Rivera

Carlos Rivera

Carlos Rivera

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?

4 of 4

Cyclic groups.
Describe.
Prove.
Reflect.
Permutation Groups.
Describe.
Prove.
Reflect.
Cosets.
Describe.
Prove.
Reflect.
Isomorphisms.
Describe.
Prove.
Reflect.