MAT 534 Course Webpage Algebra I Fall 2022

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.

Text