Instructor: Mladen Bestvina
Office: JWB 210
Office hours: Wednesdays 9:30-10:30 or by appointment. I'll also
hang around after class for any brief questions.
Text: Michael Artin: Algebra, 2nd edition
Note: For the most part I will follow the book pretty
closely but there will be some exceptions.
Meets: MWF
12:55PM-01:45PM at Union 323
Midterms: Sept 26 and
Nov 7.
Final: Tuesday, December 13,
2022, 1:00 – 3:00 pm
Prerequisites: 'C' or better in MATH 3210 AND (MATH 3220
OR MATH 4400). More informally, you should be comfortable
reading and writing proofs and with linear algebra as in MATH
2270.
Content: Chapter 1 of Artin is mostly a review. We'll
cover Chapters 2-7, and some of 8 if the time allows. The main
topics are introduction to group theory and abstract linear
algebra. They come together in the fundamental example of matrix
groups.
Grading: The final grade is based on homework, two
midterms and the final, each contributing 1/3 of the grade.
Homework: It will be assigned weekly, along with the reading from Artin for the week. You are encouraged to work in groups, but what you write should be your own work and you should list the other people in your group. I strongly encourage you to write the homework in latex. It will be due every week on Mondays at 9 am and you should turn it in through canvas. Late homework is not accepted but the lowest two scores are dropped from the count. You should read the assigned reading for the week before the corresponding lecture.
| Homework | ||
|---|---|---|
| Problems to turn in: | Due date: | Assigned reading: |
| Artin 1.1.7, 1.1.13, 1.2.6, 1.4.3, 1.M7. Bonus problem:
1.M11. Make sure you can do (but don't turn in): 1.1.2, 1.1.5, 1.2.2, 1.2.5, 1.2.7, 1.5.1 solutions |
Monday 8/29 | Artin 1.1-1.2, 1.5, permutations and determinants
handout |
| HW2 solutions | Tuesday 9/6 |
Artin 2.1-2.4, 2.9 |
| 2.5.1,2.5.3,2.5.6 for n=2, 2.6.1, 2.6.8. Extra credit
2.6.3 solutions |
Monday 9/12 |
Artin 2.5,2.6, also read 2.7 |
| 2.8.6, 2.8.9, 2.11.4, 2.12.2, 2.12.4 Extra credit 2.M14.
solutions |
Monday 9/19 |
Artin 2.8, 2.11, 2.12 |
| 3.1.2, 3.1.4, 3.1.5, 3.1.6, 3.2.2 Extra
credit: 3.1.11 (hint: x2+1=0 has no solutions
in F3 and F7 but does in F5)
solutions |
Monday 9/26 |
Artin 3.1, 3.2, 3.3 (note that there is an
odd discrepancy between section and homework numbering) |
| Midterm 1 on Sept 26. Covers Chapters
1 and 2. For half credit, by Friday bring corrected solutions to any parts a,b,i etc for which you didn't get full credit (2 points) |
||
| 3.3.2, 3.3.8, 3.4.1, 3.4.2, 3.4.3 Extra
credit 3.M2 solutions |
Monday 10/3 |
Artin 3.4, 3.5 |
| 3.5.2, 4.1.4 (hint: If A has this form show
X is in Im(A)), 4.2.1, 4.2.3, 4.3.3 Extra credit: 4.M5. solutions |
Monday 10/17 |
Artin 3.6, 4.1, 4.2, 4.3 |
| 4.4.2(a), 4.4.4, 4.5.1,4.5.2,4.5.3,4.6.4
Extra credit 4.6.8. solutions |
Monday 10/24 |
Artin 4.4, 4.5, 4.6 |
| 4.7.1, 4.7.2, 4.7.3, 4.7.7, 4.7.8 (hint:
show that a Jordan block is similar to its transpose).
Extra credit: 4.7.4 solutions |
Monday 10/31 |
Artin 4.7, 5.1, 5.2 (homework is from 4.7
only) |
| 8.5.1(b), 8.5.1(c), (for 8.5.1 you may
assume V=Rn with standard inner product)
8.5.4(a), 8.6.2, 8.6.3. Extra credit: 8.5.1(a) Hint: don't
use the cosine formula, consider (u+tv)2≥ 0,
write it as a quadratic function in t and note that
discriminant must be ≤ 0 (but explain why). solutions |
Monday 11/7 |
Artin 5.4, 8.1-8.6 (read this lightly, I am
covering only a small part of this) |
| Midterm 2 on Nov 7. Covers Chapters 3 and 4. | ||
| 6.7.1, 6.7.7, 6.7.10(a), 6.9.2 (here they
mean find the sizes of orbits), 6.9.5. Extra credit
6.7.10(b). solutions |
Monday 11/14 |
Artin 6.7, 6.9, 6.12 |
| 6.11.3, 6.11.4, 6.11.5, 6.11.7(a), 6.11.9.
Extra credit: 6.M7(b),(c). solutions |
Monday 11/21 |
Artin 6.11, 6.12, 7.1 |
| 7.1.1, 7.1.2, 7.2.3, 7.3.2., 7.4.2 (hint: a
normal subgroup of S5 will intersect A5
in its normal subgroup). Extra credit: 7.2.13. (hint: 1. N
is cyclic, 2. Aut(N) is isomorphic to Z/4, 3. Conjugation
action of G on N is by automorphisms of N, 4. Every
element of G has odd order. Justify all these statements
and put them together to give a proof.) solutions |
Monday 11/28 |
Artin 7.2-7.5, 7.7 |
| 7.7.3, 7.7.4(a), 7.7.4(b), 7.7.5, 7.7.9(a).
Extra credit: 7.6 (hint: multiplication by 2 is an
automorphism of Z/7 of order 3) solutions |
Monday 12/5 |
Artin 7.7 |
| Final:
Tuesday, December 13, 1:00 – 3:00 pm in our usual room
Union 323 half will be on Chapters 1-4, and the rest on Chapters 5-8 (parts we covered) |
||
You can contact me by email.
Required
statements
The University of Utah values the
safety of all campus community members. To report suspicious
activity or to request a courtesy escort, call campus police at
801-585-COPS (801-585-2677). You will receive important emergency
alerts and safety messages regarding campus safety via text
message. For more information regarding safety and to view
available training resources, including helpful videos,
visit safeu.utah.edu.
Title IX makes it clear that violence and harassment based on sex and gender (which includes sexual orientation and gender identity/expression) is a Civil Rights offense subject to the same kinds of accountability and the same kinds of support applied to offenses against other protected categories such as race, national origin, color, religion, age, status as a person with a disability, veteran¹s status or genetic information. If you or someone you know has been harassed or assaulted on the basis of your sex, including sexual orientation or gender identity/expression, you are encouraged to report it to the University’s Title IX Coordinator; Director, Office of Equal Opportunity and Affirmative Action, 135 Park Building, 801-581-8365, or to the Office of the Dean of Students, 270 Union Building, 801-581-7066. For support and confidential consultation, contact the Center for Student Wellness, 426 SSB, 801-581-7776. To report to police, contact the Department of Public Safety, 801-585-2677(COPS).
Accommodation: The University of Utah seeks to provide
equal access to its programs, services and activities for people
with disabilities. If you will need accommodations in the class,
reasonable prior notice needs to be given to the Center for
Disability Services (CDS), 162 Olpin Union Building, 581- 5020
(V/TDD). CDS will work with you and me to make arrangements for
accommodations. All information in this course can be made
available in alternative format with prior notification to CDS.