### MAT 534 Course Webpage Algebra I

Fall 2022

• Problem sets
• Exams
• Schedule

• Course Announcements
• Course Description
• Prerequisites
• Text
• Lectures
• Hand-backs
• Disability Support Services
• Critical Incident Management

• Course Announcements Announcements about the course will be posted here. Please check the site regularly for announcements (which will also be given in lecture). In addition to the class webpage, we will use Brightspace for course management.

• Following is a long list of categories, functors, and adjoint pairs of functors that come up regularly in algebra, sheaf theory, and topology: (pdf).
• I wrote up the proof of the rational canonical form from the lecture on Monday, October 18 (pdf). These notes also include enough of the language of rings and modules to explain the steps in the proof (primary decomposition and cyclic primary decomposition) in the language of rings and modules (the subject for the remainder of the semester).
• Since we have now proved that all finite groups have finite permutation representations and finite matrix representations, the question came up whether finitely presented groups also have (faithful) finite matrix representations, i.e., are they "linear"? There are examples of finitely presented groups that are not linear, cf. the answers to the following MathOverflow (MO) question. Of course many such groups are linear, e.g., Bigelow proved that the braid groups are linear: Braid groups are linear. In many important cases, e.g., mapping class groups, it is still not known whether these groups are linear.
• There are several other proofs that the alternating groups on at least 5 elements are simple beyond the proof from lecture. There is a lovely explanation of many different proofs by Prof. Keith Conrad at the following link: (pdf).
• The question came up in lecture whether two non-isomorphic finite groups could have isomorphic subgroup lattices, even when we "decorate" the subgroup lattice by information such as orders. In fact, these do exist, as proved in the answer to the following question from MathStackExchange (MSE).
• Here are some notes from an earlier course (MAT 312) that includes a discussion of Burnside's Formula for the number of orbits of a finite group acting on a finite set (pdf).
• Here are some notes on Jordan normal form (pdf, dvi, ps).
• I wrote up some notes on the spectral theorem for self-adjoint operators on a finite dimensional inner product space (pdf, dvi, ps).

Course Description The description in the graduate bulletin: Groups: normal subgroups, quotient groups, Lagrange's theorem, class formula, finite p-groups and solvable groups, Sylow's theorems, finitely generated abelian groups. Rings and modules: subrings, fields, prime and maximal ideals, quotient rings, ID's, PID's, UFD's, polynomial rings, field of fractions, the Wedderburn theorem, Hilbert basis theorem, finitely generated modules over a PID. Vector spaces: basis, linear maps and matrices, dual spaces, determinants, eigen values and vectors, inner products, spectral theorem for normal operators.

Prerequisites For graduate mathematics students there is no prerequisite. All other students should consult with the instructor regarding background.

David S. Dummitt and Richard M. Foote, Abstract algebra, 3rd ed. (ISBN: 978-0-471-43334-7) available at Shop Red West. A copy of the textbook will be on reserve in the Mathematics, Physics and Astronomy Library Also you are encouraged to look for new or used copies of the book through other retailers or through your fellow graduate students. This is the same textbook used in the past, and I am sure there are good bargains to be had.

Lectures The instructor for this course is Jason Starr. All instruction will occur in lectures. There are assigned readings in the course schedule which are to be completed before lecture. During lecture the instructor and the students will discuss the material in those readings, there will be exercises to practice the material, etc. For the lectures to be effective, you must complete the assigned reading from the syllabus before lecture.

Lecture is held Tuesdays and Thursdays, 1:15 — 2:35 PM in Physics P130.

Here is a link to the current office hours.

 Midterm I 20% Midterm II 20% Final Exam 35% Problem Sets/Class Participation 25%
Regarding problem sets and class participation, this fraction of the grade will mostly be determined by scores on the problem sets. However, especially if there are problems with low attendance, some fraction may be determined by attendance, class participation, performance on pop quizzes, etc. (in that case, the distribution of the 25% will probably be 20% problem sets, 5% class participation).

Letter grades are based on total class percentage, following the weights above. Students who earn percentages in the following ranges will earn (at least) the listed letter grade.
A 95-100; A- 90-94; B+ 86-89; B 83-85; C+ 75-78; C 71-74; C- 67-70; D+ 62-66; D 58-61; F 0-57

Hand-backs
Graded problem sets and exams will be handed back in lecture. If you cannot attend the lecture in which a problem set or exam is handed back, it is your responsibility to contact your instructor and arrange a time to pick up the work (typically in office hours).

You are responsible for collecting any graded work by the end of the semester. After the end of the semester, the instructor is no longer responsible for returning your graded work.

### Disability Support Services

If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the Student Accessibility Support Center. For procedures and information go to the following website: https://ehs.stonybrook.edu//programs/fire-safety/emergency-evacuation/evacuation-guide-disabilities and search Fire Safety and Evacuation and Disabilities.

Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html

### Critical Incident Management

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Student Conduct and Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures. Further information about most academic matters can be found in the Undergraduate Bulletin, the Undergraduate Class Schedule, and the Faculty-Employee Handbook.

Jason Starr
4-108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 11794-3651
Phone: 631-632-8270
Fax: 631-632-7631
Jason Starr