Math 402 A: Introduction to Modern Algebra
Autumn 2016

Instructor: Alexandru Chirvasitu

Class times and location:
Office hours: Wednesday 08 - 09 or by appointment
Email: chirva AT uw.edu

TA: Li Li
Office hours: Tuesday 3:30 - 5
Email: lil37 AT uw.edu

The course is an introduction to abstract algebra using as its vehicle the study of mathematical gadgets known as groups. These are abstract constructs aimed at capturing the structure and properties of symmetry.

Objects in nature often exhibit symmetry of various kinds, meaning that the object in question is "the same" upon performing certain operations on it. Think about the rotational symmetry of a starfish or the bilateral symmetry of human bodies. The former means that if you do a fifth of a full rotation on the starfish you can't tell the difference, and the latter means that your left hand is the mirror image of your right hand, etc. These operations that leave the object invariant come pre-packed with a lot of structure: you can compose them (turn the starfish around twice or three times rather than just once), invert them (turn clockwise ratherthan counterclockwise), and so on.

In math we like to aggregate the items we are interested in (the symmetries) into a single structure, and study that in the abstract, with the goal of then applying the resulting insights to all of the examples provided either by nature or by our own minds. For each given object, its symmetries form exactly what we are going to call a group; these are the main characters of the course.

Think of the course as having a twofold goal:
• First, we'll be studying groups for their intrinsic interest; the theory behind this type of structure is very rich, and depending on how mathematically minded you are you might find it fascinating in its own right.
• Secondly, you'll get to see in action the general principle stated above of how math functions and what it's for, and hence get acquainted through a specific example with what mathematicians do on a daily basis: take something concrete (symmetries), isolate abstract structure and properties that you feel capture its nature (ability to compose or invert or whatever), organize those into a piece of abstract math (a group), and then study that in order to gain insight that feeds back into the concrete realization you started out with.

Textbook

We're using Dan Saracino's Abstract Algebra (2nd edition). The textbook is absolutely necessary, both for the reading and in order to do the assignments.

It's essential that you do the reading. I won't have time to go over every relevant example in class, and you'll need a good grasp of the material in order to do the homework. In fact, I encourage you to regard the reading as part of your homework.

The phrase 'Section x', page numbers and result numbering (such as 'Theorem a.b') always refer to our textbook.

Due date Assignment Remarks
1 Fri Sept 30 Sections 0 and 7 This is review for sets and functions.
The material constitutes the foundational framework or
what we're doing, so make sure you're comfortable with it.
2 Mon Oct 03 Section 1
3 Wed Oct 05 Section 2
4 Fri Oct 07 Section 3
5 Mon Oct 10 Section 4
before Theorem 4.2
6 Wed Oct 12 finish Section 4
Section 5, stop at 9. (page 47)
7 Fri Oct 14 finish Section 5
Section 6
8 Mon Oct 17 Section 8;
stop after Theorem 8.4
9 Wed Oct 19 finish Section 8 Eeverything up to here
10 Mon Oct 24 study for the midterm; treat this
meeting as extended office hours / Q&A for the test
11 Fri Oct 28 Section 9
12 Mon Oct 31 Section 10;
stop before Theorem 10.4
13 Wed Nov 02 finish Section 10
14 Fri Nov 04 Section 11;
stop after Corollary 11.5
15 Mon Nov 07 finish Section 11
16 Wed Nov 09 Section 12;
stop after Theorem 12.1
There will be no class on either Nov 11
or the entire week of Nov 14.
17 Mon Nov 21 finish Section 12
18 Wed Nov 23 Section 13;
stop after 5. (page 125)
19 Mon Nov 28 continue Section 13;
stop before Theorem 13.5
20 Wed Nov 30 finish Section 13
21 Fri Dec 02 Section 14;
stop before Corollary 14.6
22 Mon Dec 05 finish Section 14
23 Wed Dec 07 study for the final; treat this
meeting as extended office hours / Q&A for the test,
24 Fri Dec 09 no meeting in class; extra test-review
office hours held by our TA
2:30 - 4:30 in Padelford C-430

Supplementary material

On occasion, I'll post extra notes, comments, etc. in this space.
• A complete proof of the main result from the Friday, Nov 04 lecture.

Homework

I am denoting the problems by x.y, as the textbook does (to mean problem y from Section x). Whenever you solve a problem, you can cite any preceding problem or theorem in your solution.

You have to turn in the assignment at the beginning of class, before the lecture. No late homework for any reason, but I will drop the lowest score.

Because of time constraints your TA will grade a couple of problems (tops) for correctness and the rest for completeness. I won't be telling you in advance which problems are graded for correctness though..

Due date Assignment Remarks
1 Wed Oct 05 0.4, 0.5, 0.14, 0.16
7.2, 7.4, 7.6, 7.9
1.2 - 1.5, 1.7 - 1.9
A dash '-' means the whole range,
so 1.2 - 1.5 means four problems.
2 Wed Oct 12 2.1, 2.3, 2.8, 2.10
3.1, 3.3, 3.11, 3.13
4.8, 4.9, 4.16, 4.24
3 Wed Oct 19 5.9, 5.10, 5.12, 5.14, 5.22
6.3, 6.5
8.1, 8.4, 8.12, 8.16, 8.23
Your homework problems up to here
are midterm prep.
No hw on Oct 26
because of the midterm
4 Wed Nov 02 9.1, 9.3, 9.4, 9.11, 9.16, 9.18
10.3, 10.7, 10.11, 10.13, 10.24, 10.26
5 Wed Nov 09 11.1, 11.2, 11.8, 11.16, 11.18, 11.23
12.1, 12.2, 12.3, 12.5
We are not meeting
the week of Nov 14.
6 Wed Nov 23 12.10, 12.11, 12.14, 12.23, 12.30, 12.35
13.1, 13.2, 13.7
7 Wed Nov 30 13.11, 13.12, 13.14, 13.15,
13.19, 13.25, 13.26, 13.27
8 Wed Dec 07 14.2, 14.3, 14.4, 14.7
14.9, 14.12, 14.13

Midterm

It's happening in class (usual time and place):
• Wednesday, October 26 2016, 09:30 - 10:20, room 008 Anderson Hall
The test is open book.

Solutions are now available.

Final

The date and time are set by the department and I cannot change it. Early / late exams are not an option, so please plan accordingly.
• Wednesday, December 14 2016, 08:30 - 10:20, room 008 Anderson Hall
Both the midterm and the final are open book exams. You get to bring your book, class notes, past homework, whatever (just do not collaborate with anyone during the tests).

• Homework : 20%
• Midterm : 35%
• Final : 45%

As mentioned above, we'll drop the lowest homework score.