Math 100A/B/C is a rigorous three-quarter introduction to the methods and basic structures of higher algebra. 100A will focus on group theory. Topics include: groups, subgroups and factor (quotient) groups, homomorphisms, permutation groups, matrix groups, and some advanced topics as time permits (e.g., the Sylow theorems).

UCSD also offers a two-quarter algebra sequence, Math 103A/B (offered fall/winter, winter/spring, and in summer sessions). Between the two, Math 100 offers a greater emphasis on concepts and mathematical rigor, as well as some advanced topics not covered in Math 103 (e.g., Galois theory). Math 100 is recommended for students planning further study in pure mathematics, while Math 103 is recommended for most other students; however, any student interested in exploring these topics at a deeper level is welcome!

Students may not receive credit for both Math 100A and Math 103A. While 100A is a valid prerequisite for 103B, 103A is **not** a valid prerequisite for 100B without permission of the 100B instructor.
(If you took 103A and want to audit 100A, contact an instructor for assistance.)

During fall 2022, there will be two parallel lectures of Math 100A; these will be distinct but highly coordinated, and this syllabus applies equally to both lectures. Both lectures are equally valid prerequisites for 100B.

Most communication about the course will take place on the Zulip platform; in particular, all Zoom links will be posted there. To get the link for Zulip, complete the Checklist survey on Canvas.

All course activities will be available for remote participation **except** for the two midterms and the final exam. See the course policies below for the allowed accommodations.

If you are an enrolled UCSD student and wish to audit the course, contact an instructor to be added to Canvas.

**Instructors:**

- Be'eri Greenfeld, bgreenfeld [at] ucsd [etcetera].
- Kiran Kedlaya, kedlaya [at] ucsd [etcetera].

**TAs:**

- Bryan Hu, brhu [at] ucsd [etcetera].
- Finn McGlade, fmcglade [at] ucsd [etcetera].

**Lectures:** Kedlaya: MWF 9-9:50am in Peterson 102; Greenfeld: MWF 10-10:50am in Warren Lecture Hall 2205.
Please attend only your assigned lecture; you may watch recordings of both lectures in Canvas.
Masks will be required in lecture as long as campus guidelines require it.

**Sections:** McGlade: Wed 6-6:50pm or 7-7:50pm in Warren Lecture Hall 2112; Hu: Thu 8-8:50am or 9-9:50am in APM B402A. You may attend any section (of either lecture) as long as you make space for all students assigned to that section.

**Office hours:** These will be moved or augmented some weeks; watch Zulip for announcements.

- Instructor (Greenfeld): MWF 11-11:50am (in person, hybrid upon request). Location: APM 6434.
- Instructor (Kedlaya): Wed 12-1pm (in person, hybrid upon request), Thu 8-9pm (remote), Fri 1-2pm (in person, hybrid upon request). Location: APM 7202.
- TA (Hu): Thu 10-11am, Thu 3:30-5:30pm, Fri 2-3pm (all in person, hybrid upon request). Location: HSS 5024.
- TA (McGlade): Wed 3-5pm (remote), Fri 3-5pm (in person). Location: HSS 4012.

**Text:**
Algebra by Michael Artin, second edition (required). The hardcover, softcover, and eBook versions of the text are all interchangeable. For 100A, only chapters 1-7 will be used; the same text will be used for 100B and 100C this year.

**Prerequisites:**
Math 31CH or Math 109 or consent of instructor.

**Homework:** Weekly assignments, submitted via Gradescope, due Fridays at 11:59pm.
There will be no homework due in week 9 (Thanksgiving); homeworks due the week of a midterm will be shorter than usual, but will count equally in the course grade. Homework due in week 1 is not for credit.
No extensions will be granted; however, there will be opportunities to submit revisions to bring selected assignments up to 80% total credit.

**Midterm exams:** Friday, October 14 and Wednesday, November 9 in class.
No makeup exams will be given, nor remote exams except as provided for by course policy (see below).
However, for each midterm there will be one opportunity to submit revised work for up to 90% total credit.

**Final exam:** Kedlaya: Wednesday, December 7, 8-11am, Peterson 102;
Greenfeld: Friday, December 9, 8-11am, Warren Lecture Hall 2205.
You may request **in advance** to switch to the exam time for the other lecture;
otherwise, no makeup exam will be given,
nor a remote exam except as provided for by course policy (see below).
See also UCSD exam policies.

**Grading:** 45% homework (including revisions, not counting week 1); 15% each midterm (including revisions);
25% final exam. Each assignment (whether long or short) will be weighted equally; the lowest homework assignment (after revisions) will be dropped.
All grades will appear in Gradescope, not Canvas.

The conversion of raw percentages into letter grades will be made in order to maintain a grade distribution comparable with historical averages for this course. However, the following minima are guaranteed:

Percentage | 97 | 93 | 90 | 87 | 83 | 80 | 77 | 73 | 70 |

Minimum grade | A+ | A | A- | B+ | B | B- | C+ | C | C- |

Notwithstanding the above, to receive a passing grade, you must fulfill the following conditions.

- You must take the final exam, at the scheduled time and place (unless granted an accommodation according to course policies), and receive a passing grade. If you are forced to miss the final because of a COVID-19 quarantine or other medical issue, please request an Incomplete (see below).
- You must not be found in violation of UCSD's academic integrity or harassment policies.

No extensions will be given for homework assignments, but the lowest homework assignment will be dropped. In addition, there will be an opportunity at the end of the term to submit revisions for total credit of up to 80% on each assignment.

Students who join the course from the waitlist, or who transfer between the two lectures, will be responsible for all homework from the start of the term. If you are on the waitlist, contact a member of course staff in order to get access to Gradescope and Zulip.

At the top of each homework assignment, you must specify all outside resources that you consulted, or write "None" if none were used. YouAll exams will be closed-book: no outside materials may be consulted. This includes the textbook, lecture notes, the Internet, and anyone other than the exam proctor(s). We reserve the right to:

- require students to produce their UCSD student ID cards for admission to the exam room;
- assign seating before or during the exam;
- make video recordings for the purposes of monitoring academic integrity.

Exam accommodations will be made only in the following cases.

- For disability accommodations (including those related to COVID), please contact the Office for Students with Disabilities. If your request is approved, we will be notified automatically.
- For athletic accommodations, please have a cognizant representative of the Department of Athletics contact the instructors.
- For religious accommodations consistent with UCSD policy, please contact the instructor.

No makeup exams will be given. A missed midterm exam will score 0, subject to revision (see above).

A request for an Incomplete grade will only be granted in accordance with UCSD policies. In particular, you must be on track to receive a passing grade based on your submitted homework and midterm results (including revisions). To convert an incomplete into a final grade, you must provide to the instructor proper documentation of the circumstances leading to the Incomplete, and arrange with the instructor to complete all outstanding course requirements no later than the end of the subsequent quarter.

Violations of UCSD's academic integrity policies (cheating, plagiarism, etc.) will be handled by the instructor using UCSD administrative measures. In addition, the instructor reserves the right to assign a 0 score to any homework or exam affected by a violation. If you suspect a violation, please bring it to the attention of the instructor and/or TA immediately.

- Friday, September 23: overview of the syllabus. 9am lecture over Zoom.
- Monday, September 26: laws of composition (2.1); definition of groups, examples (2.2).
- Wednesday, September 28: permutations (1.5); cancellative law (2.2).
- Friday, September 30: subgroups of the additive group of integers (2.3).
- Monday, October 3: cyclic groups, Klein four group, quaternion group (2.4).
- Wednesday, October 5: homomorphisms and isomorphisms (2.5, 2.6).
- Friday, October 7: kernel and image of a homomorphism, conjugation (2.6).
- Monday, October 10: equivalence relations (2.7).
- Wednesday, October 12: cosets, Lagrange's theorem (2.8).
- Friday, October 14: midterm 1 in class.
- Monday, October 17: Lagrange's theorem, normal subgroups (2.5, 2.8).
- Wednesday, October 19: quotient groups (2.12).
- Friday, October 21: correspondence theorem (2.10); first isomorphism theorem (2.12).
- Monday, October 24: product groups (2.11).
- Wednesday, October 26: isometries of Euclidean spaces (6.2).
- Friday, October 28: isometries and orthogonal matrices (6.2).
- Monday, October 31: classification of finite subgroups of SO(2) (6.4).
- Wednesday, November 2: dihedral groups; classification of finite groups of orthogonal matrices (6.4).
- Friday, November 4: group operations, stabilizers (6.7, 6.8).
- Monday, November 7: TBA. No 9am lecture; watch the recording of the 10am lecture instead.
- Wednesday, November 9: midterm 2 in class.
- Friday, November 11: no lecture (Veterans Day).
- Monday, November 14: the operation on cosets, counting formula (6.8, 6.9).
- Wednesday, November 16: operations on subsets, finite subgroups of the rotation group (6.10, 6.12).
- Friday, November 18: finite subgroups of the rotation group (6.10, 6.12).
- Monday, November 21: Cayley's theorem, the class equation (7.1, 7.2).
- Wednesday, November 23: p-groups, class equation of the icosahedral group (7.3, 7.4).
- Friday, November 25: no lecture (Thanksgiving).
- Monday, November 28: simplicity of alternating groups (7.5).
- Wednesday, November 30: Sylow p-subgroups, statement of the Sylow theorems (7.7).
- Friday, December 2: proof of the first Sylow theorem; groups of order 12 (7.7, 7.8).