Description of the course

The course covers the basic theory of groups and rings. 

Roughly the first two thirds of the semester will be spent on groups.  We will start with examples of how groups arise naturally in different areas, often as groups of symmetries (automorphisms). Some examples: Permutation groups, General linear groups, subgroups of general linear groups arising from linear algebra, dihedral groups.

After introducing some examples, we begin to develop the theory. It is important to not just consider groups in isolation, but to study their relation to each other. For this purpose we introduce the concepts of homomorphism, isomorphism, kernel, normal subgroup, quotient group; the three isomorphism theorems.

One of the main motivations for inventing groups is that they provide a good framework for studying symmetries. The relationship between groups and symmetries is encoded in the concept of a group action. Group actions are the reason that groups play a central role in other branches of mathematics, and also outside of mathematics. But group actions are useful also for the study of groups themselves: one can learn a lot about the structure of groups by investigating the way they act on other objects. Important examples: Action of G on itself by left multiplication and by conjugation. 

Using what we learned about group actions and the Isomorphism theorems, we will prove Sylow's theorems and some related results about groups whose order is a prime power. Sylow's theorems are the first deep results we encounter about the structure of finite groups. We can use them to classify groups of certain orders, and to show that there are no simple groups of certain orders. For example, we will check that a simple group of order at most 100 necessarily has order 60 (in fact it is isomorphic to the alternating group A_5).

The Sylow theorems are the culmination of our study of groups. After that, we will move on to the second part, which is concerned with rings. Here is the outline of the second part:

Basic definitions about rings and fields, including: Ideals, prime ideals, maximal ideals, integral domains. 

Important Classes of Rings: Euclidean Domains, Principal Ideal Domains (PID), Unique Factorization Domains (UFD).

Important Classes of ideals: Maximal ideals, prime ideals, principal ideals, finitely generated ideals.

Important Classes of elements: Units, irreducible elements, prime elements, nilpotent elements and zero-divisors.

Examples of rings and their applications: Chinese Remainder Theorem, Polynomial Rings, irreducibility criteria.