Schedule

I asked students to take a short diagnostic examination. This exam will NOT affect your grades in any way. Though an unexciting way to start the semester, this is the best way for me to gague student background and thought processes.

Students are required to read (and keep notes on) our syllabus before the next meeting. If you have any questions, please bring them to that lecture.

I gave solutions to the diagnostic exam questions during this lecture. This should give students an idea of how I expect problems to be solved and proofs to be written throughout the semester.

Students should write clean solutions to the problems we didn't get to.

Some of these will be presented during a future lecture.

• Solve the following exercises from textbook section 1.4.

  Sets and Logic

  1 – 3, 4 – 8 (evens), 11, 14 – 15. (Solve at least three).

  Functions

  19, 22 – 24. (Solve at least two).

  Relations

  21, 25 – 26, 28. (Solve at least three).

See my notes for a brief refresher of some elementary number theory. Chapter 2 from the textbook also covers similar content.

We looked at the integers with addition to abstract some properties. We then considered the symmetries of the triangle, and which of these were still satisfied.

• Read chapter 2 from the textbook. This is about 10 pages of reading, but it should mostly be review.
• For a future lecture, attempt the following exercises from textbook section 2.4:

  Induction

  3, 4 (also express the left side in sigma notation), 5.

• Abstract definition
• General dihedral groups of order \( 2n \).
• Common Groups (\( (\mathbb{Z}, +), (\mathbb{Q}, +), (\mathbb{R}, +), (\mathbb{C}, +) \))

• Read textbook section 3.1.

See this document for presenters and their problems.

Update: Thanks to everyone who presented!

In the time remaining after presentations, we started a proof that \( M_{2, 2}(\mathbb{R}) \) is a group under matrix addition.

Solve these exercises.

Be sure that it is typed up in LaTeX (it will not be accepted if written by hand). See my LaTeX notes for a hand with this (including a template LaTeX file for writing homework). Also feel free to come to office hours!

• \( M_{2, 2}(\mathbb{R}) \) under addition [yes] and multiplication [no]
• The unit circle as a group
• \( \mathbb{Z} / n\mathbb{Z} \) under addition [yes] and multiplication [no]
• \( U(n) \), the units mod \( n \), under addition [no] and multiplication [yes]
• Functions of a set \( S \): injections [no], surjections [no], and bijections [yes]

We proved that the identity element of a group is uniquely determined.

• Read and take notes on textbook section 3.2.

I'm traveling for a conference this weekend. Students agreed to work together on this problem set while I'm away. I suggest you meet at the usual class time to discuss the problems together.

You only need to turn in one copy of the assignment per group.

I'm still traveling (this time towards Tenessee). Students agreed to work together on this problem set while I'm away. I suggest you meet at the usual class time to discuss the problems.

You only need to turn in one copy of the assignment per group.

For all \( n \in \mathbb{N} \) and all \( g \in G \), we denote the \( n \)-fold products \( g \cdot g \cdot \cdots \cdot g = g^n \), and \( g^{-1} \cdot g^{-1} \cdot \cdots \cdot g^{-1} = g^{-n} \).

A subgroup of group \( (G, \cdot) \) is a group \( (H, \cdot_H) \) where \( H \subseteq G \), and \( \cdot_H \) is \( \cdot \) restricted to \( H \). In particular, a subgroup is a subset of a group which forms a group under the restricted operation.

After defining subgroups, we looked at a few examples.

Solve this exercise.

Solve these exercises.

Turn in the problem set you worked on together while I was away. You only need to turn in one copy of the assignment per group.

We discussed more examples of subgroups of a group. We then gave a proof of the Three Step Subgroup Test.

Solve (at least one of) these exercises.

NOTE: I've changed the due date of homework 03 to give you a few more days (that makes at least two weeks that the assignment will have been available).

See this document for presenters and their problems.

Update: Thank you to all the presenters!

We proved that the center of a group is a subgroup. We discussed (at some length) how to work with group operations and think about subgroups.

The order of an element \( g \in G \) is \[ \vert g\vert = \inf\{k \in \mathbb{Z}_{>0} : g^k = e\} . \] We noted that this set can be empty (hence \( \inf \) instead of \( \min \), allowing \( \vert g \vert = \infty \)). We then proceeded to work through several examples.

Solve these exercises.

Note: I plan to hand these back on Friday, during the review session. This is among the best-laid plans of mice and men… (Update: I made it happen!)

Students should bring questions to the lecture. Homeworks and the like are a good source of questions.

Update: Thank you for bringing your questions!

During the usual class time. Following the exam, please watch this video to prepare for future lectures.

Update: Exam 1 has been graded and returned. Students should feel free to meet with me on office hours to discuss the problems and their solutions.

Definition

A group \( G \) is cyclic when there is a \( g \in G \) such that for all \( x \in G \) we have \( x = g^k \) for some \( k \in \mathbb{Z} \). The element \( g \) is called a generator for \( G \).

Examples

Let \( n \in \mathbb{Z} \).

1. \( n\mathbb{Z} \) is cyclic.
2. \( \mathbb{Z} / n\mathbb{Z} \) is a cyclic group.
3. The set of rotations of a regular \( n \)-gon is a cyclic subgroup of \( D_n \).
4. The set of \( 2 \times 2 \) matrices with integer entries and having the form \( \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \)
5. The units group \( U(p) \) for prime \( p \).

Non-Examples

None of the following are cyclic.

1. The dihedral group \( D_n \) for \( n \geq 3 \).
2. The set \( S_n \) of self-bijections of \( \{1, 2, \dots, n\} \) under function compositions. This is called the symmetric group on \( n \) letters, and features prominently in group theory.
3. The special linear group \( \mathrm{SL}_n(\mathbb{Z}) \).

• Solve (at least one of) these exercises.
• Read about and take notes on cyclic groups. Focus your attention on section 4.1. You can read the other sections if you would like to see some of their cool applications.

Proved several propositions concerning cyclic groups, their subgroups, and conditions on whether or not an element can generate the whole group.

• Read about and take notes on cyclic groups (if you haven't already).
• Solve (at least one of) these exercises.

Finished proving that if \( G = \langle g \rangle \) is finite of order \( n \), then \( |g^k| = \frac{n}{\gcd(n,k)} \).

Refresher on functions and the terminology surrounding them. This included reminders about the definitions and properties of injective, surjective, and bijective functions.

Having only a few minutes left, I ended with a fun aside about right inverses and the Axiom of Choice. Students agreed that mathematical logic is "the best thing ever".

• Refresh yourself on functions.
• Solve (at least one of) these exercises.

The symmetric group on set \( X \) is \( S_X \), the set of bijective functions \( X \to X \). Such bijective functions are permutations of \( X \). When \( X = \{1, 2, \dots, n\} \), we abbreviate to \( S_n \).

We did the following.

• looked at \( S_1 \), \( S_2 \), and \( S_3 \) in depth,
• learned about array notation for permutations, and
• began proving that \( S_X \) is indeed a group (completed closure and associativity).

• Read about permutations groups (section 5.1).
• Solve (at least one of) these exercises.

• Finished our proof that \( S_X \) is a group.
• Discussed examples and basic properties of \( S_n \).

• Continue to read about permutations groups (section 5.1).
• Solve (at least one of) these exercises.

• Cycle representation.
• Composition of cycles.

• Read more about permutations groups (section 5.1). Pay particular attention to their discussion of cycles and transpositions.
• Solve (at least one of) these exercises.

• Discussed different ways to express permutations.
• Outline a proof that there exists (an essentially unique) decomposition of every element of \( S_n \) as a product of disjoint cycles.

• Solve (at least one of) these exercises.

See this document for presenters and their problems.

Update: Thanks to all who presented!

• Began the proof of the essential uniqueness of cycle decomposition.

• Continue to read about permutations groups (section 5.1).
• Finish the proof that disjoint cycles commute.
• Have a restful fall break!

• Illustrated the existence portion of the proof with an example.
• The full proof appears in a video (linked at a later lecture date).

• Defined transpositions.
• Showed via counterexample that transpositions do NOT allow us to decompose elements uniquely.
• A video will prove that it is always possible to express a permutation as a product of transpositions.
• Another video will prove that the number of transpositions used to express \( \sigma \) has the same parity (but not necessarily the same number) regardless of how we express it as a product of transpositions.

Watch and take notes on this video on cycle decompositions. Note that some details are left to you! You can also read the actual notes from the actual video.

Watch and take notes on this video on fundamental properties of transpositions. After that, do the same for this video on the parity of a permutation. Note that some details are left to you! You can also read the actual notes from the actual video.

Went through the proof that parity of transpositions is well-defined, using an example to illustrate the key ideas.

Defined cosets and calculated an initial example.

• Read about cosets and Lagrange's Theorem.
• Prove that the alternating group is a subgroup of the symmetric group.

• Email me with your availability for our oral exam 2. Plan to speak with me for 45 minutes.

  Update: I've emailed you with the planned time based on everyone's availability.

• Worked through some examples computing cosets together.
• Began a proof that some five conditions concerning cosets are logically equivalent. These conditions are incredibly important to what we do in the future…

• Solve (at least one of) these exercises.
• Read about cosets and Lagrange's Theorem.

• Proved the equivalence of five conditions regarding cosets.
• Proved that \( x \sim y \) when \( x^{-1}y \in H \) is an equivalence relation whose equivalence classes are the left cosets.
• Proved that cosets are equicardinal, i.e., \( |xH| = |yH| \) for all \( x, y \in G \).
• Proved Lagrange's Theorem: If \( H \leq G \) then \( |H| \big\mid |G| \).

• Solve (at least one of) these exercises.
• Read about cosets and Lagrange's Theorem.

• Index of a subgroup.
• Cool theorems.

• Solve (at least one of) these exercises.
• Read about cosets and Lagrange's Theorem.

Definition and invariants of various kinds.

• Read about isomorphism.
• Solve (at least one of) these exercises.

We proved the following.

• Described some "global isomorphism invariants" of groups, e.g., abelian and cyclic.
• Every cyclic group is isomorphic to either \( \mathbb{Z}/n\mathbb{Z} \) or \( \mathbb{Z} \).
• Cayley's Theorem: Every group is isomorphic to a permutation group.

• Read about product groups.
• Solve (at least one of) these exercises.

• Defined the external direct product.
• Gave some examples.
• Discussed properties.

See this document for presenters and their problems.

• Read about normal subgroups and quotient groups
• Solve (at least one of) these exercises.
• Note problem 55 in the textbook. This would be an excellent problem for you to use to test your understanding of the course so far. I won't assign it, but you might try it.

• Discussed the definition.
• Worked through multiple examples and non-examples.
• Proved some fundamental properties.

• Keep reading about normal subgroups and quotient groups
• Solve (at least one of) these exercises.

• Discussed the definition.
• Worked through multiple examples and non-examples.

• Keep reading about quotient groups
• Solve (at least one of) these exercises.

• Proved that they are, indeed, groups!
• Discussed their use and more examples.

• Solve (at least one of) these exercises.

• Discussed nested normal subgroups.
• Found counterexamples and proofs for several statements regarding nested normal subgroups.

• Definition.
• All of the examples.
• Some of the non-examples.
• Basic properties.

• Solve (at least one of) these exercises.

Discussed the homomorphism theorem, which describes the conditions under which a homomorphism "factors through" a projection to a quotient group.

• Proved the First Isomorphism Theorem.
• Discussed examples with the First Isomorphsm Theorem.
• Stated and briefly discussed the Second and Third Isomorphism Theorems.

Sketched a proof of the Fundamental Theorem of Finite Abelian Groups. This required tying a lot of propositions over the course of the semester together.

This is the last day to hand in homeworks from before meeting 39.