August 1 Math 371, second-semester Abstract Algebra, course which emphasizes on rigorous proofs. Students are assumed to be familiar with the mathematical language, and understand what constitute a correct proof.
The prerequisite of this course is rigorous linear algebra and basic group theory. These topics are covered in Math 370 and also in Math 502. See the first part Review 1 for a list of concepts that students are assumed to be familiar with, as well as some standard notations. There are also a few problems in the second problems of Review 1; we will go over (some of) them in class and lab sessions.
Since the materials in Math 371 tend to be somewhat abstract, examples are crucial. They offer a testing ground for the concepts and theorems. Having a large number of examples at hand helps you to develop a feeling of whether a statement is plausible or not. A collection of common examples of groups can be found in Review 1. You are strongly advised to go through these examples and play with them. For instance, take a groups G in the list, and pick a non-trivial subgroup H of G, you may want to figure out the center Z(G) of G, the normalizer of H in G. Later on, for a finite group G on the list, you want to figure out the Sylow-p subgroups for each prime number p which divides the order of G. In a sense the body of mathematics is a collection of (interesting) examples. The abstract concepts and theorems are useful ways for us to organize these examples.
As you must have discovered in Math 370, a course in rigorous abstract mathematics is quite different from the calculus courses we offer. In particular you cannot expect to be able to do a homework problem by looking up a similar example in the textbook. The most important part of your learning process for this course is doing the problems. It is often said that "you have not learned anything if you cannot do the problems".
The majority of homework problems are about examples of the the abstract concepts in action, so that in principle you should be able to make substantial progress if you understand the meaning of the key words. Let us illustrate this point with problem 3 of Assignment 1, which asks you to find all homomorphisms from a cyclic group of order 3 to a cyclic group of order 6. How can one approach this problem? Well, first you tell yourself that this problem can be determined in a finite amount of time: Every such homomorphism can be regarded as a function, from a set with 3 elements to a set with 6 elements, and there are 216 such functions. This is a start, and if you are persistent enough you can go through the list of 216 possibilities, and decide which ones are homomorphisms from the definition. So already you know that in principle you can do it. Then you remember that a homomorphism sends the unit element of the source group to the unit element of the unit element of the target group. This cuts down the number of possibilities to 36, and it is not too bad to sort out among the 36 candidates which ones are homomorphism. If you use other defining properties of homomorphisms, you can further cut down the number of possibilities, In the lab sessions you will be able to compare other people's solutions with your own. If you see a solution more elegant than your own, analyze it. You have the option of talking to other students in the class, but you should always try the problems yourself first! Your first approach may be awkward in retrospect, and that is all right. But NEVER just give up!The presentation in the lectures will be different from that of the textbook. You are encouraged to ask question, take notes. Then work out your notes after class and compare with the approach in the textbook.