Math 521A

Math 521A, Abstract Algebra
(2:45-4:00 MW in C-9)

Syllabus: Here is the syllabus.

Online resources:

Homework: Remember that you do not have to turn in the HW. However you should keep up with it. This is why we have target dates. You should have completed your HW by the target date and have it in a form that you would be confident relying on at the next exam. You can of course come back to it and revise it afterwards.

Assigned	Target date	Read	Exercises
9/5	9/12	Ch. 0, pp. 3-17	0.4, 10, 12, 14, 17
9/12	9/19	Ch. 0, pp. 8-22	0.8, 13, 15, 18, 19
9/19	9/26	Ch. 0, pp. 8-22	0.22, 24, 25
9/26	10/3	Ch. 0, pp. 17-22	0.16, 26, 28, 49
10/3	10/10	Ch. 0, pp. 17-22	0.48, 50, 51 1. Find the error in the following argument. Let ~ be a relation on a set S. Suppose ~ is summetric and transitive. We claim that ~ is an equivalence relation. Since ~ is already symmetric and transitive, all we need to prove is that ~ is also reflexive. Let x be an element of S. If x~y then y~x by symmetry. Now x~y and y~x imply x~x by transitivity. Hence ~ is indeed reflexive and an equivalence relation. 2. Find a relation ~ on a set S which is symmetric and transitive, but not reflexive.
10/10	10/17	Ch. 1	0.53 1. Let f:A->B and g:B->C be functions. Prove that if gf is onto, then g is onto. 2. Let S be a finite set and f:S->S a function. Prove that f is one-to-one if and only if it is onto. 3. Let S=natural numbers. Find a function S->S which is (a) one-to-one but not onto, (b) onto but not one-to-one.
10/17	10/24	Ch. 1	Let f:S->T be a function. Prove that 1. f is one-to-one if and only if there exists g:T->S such that gf=id_S, 2. f is onto if and only if there exists g:T->S such that fg=id_T.
10/24	10/31	Ch. 2	1.5-8, 14, 22
10/31	11/7	Ch. 3	2.4, 6, 14, 16, 18, 32
11/7	11/14	Ch. 3	3.6, 11, 23, 28, 44 Find a mistake in the statement in exercise 3.26. (Hint: are the elements of Z_n really integers?) If you don't see a mistake, prove that the statement is false.
11/14	11/21	Ch. 3	3.10, 12, 14, 18, 27
11/21	11/28	Ch. 4	3.15, 16, 20, 24, 32, 52
11/28	12/5	Ch. 4	4.14, 18, 22, 36, 42, 54

The problem of the fortnight: The Mathematics Department in San Diego posts a new fun problem every two weeks. If you'd like to submit a solution, give it or e-mail it to me, so that I can fax them all together to San Diego. Winners receive prizes.

The Putnam Competition: This is not strictly related to this course. The William Lowell Putnam Mathematical Competition will be held on Sat, Dec 1 this year (it is administered locally right here on campus). The exercises primarily test the ability to construct rigorous mathematical arguments to solve (difficult) unfamiliar problems and not lexical knowledge. If you like proofs and enjoy the challenge of thinking about such problems, I encourage you to sign up for the competition by contacting me soon. Registration must reach the organizers by Oct 12, so I will send it out around Oct 5. Here is an archive of past exercises and solutions. We will have a series of training sessions to prepare for the Putnam.

Exam solutions:

First hourly exam pdf
Second hourly exam pdf
Third hourly exam pdf
Final exam pdf

Useful links: The links below lead to sites with encyclopedias of math terms. You can use them to find definitions, examples, and some theorems.

Other algebra books: Here are a few other textbooks you may want to consult.

J.A. Beachy and W.D. Blair, Abstract algebra, Waveland Press. This is an undergrad textbook, whose level is closest to our text's. This was originally meant to be the textbook for our course. I've put my copy on reserve in the library. Check it out!
M. Artin, Algebra, Prentice Hall. This is a textbook for a rather sophisticated undergrad algebra course. Artin likes matrices.
I.N. Herstein, Topics in algebra, Wiley. This is a classic textbook used by many undergrad and some graduate level algebra courses. It's more sophisticated than our course, but you may still get some good ideas from it.
I.N. Herstein, Abstract algebra, Wiley. This is a lighter version of the above book. In fact, it's often referred to as "baby Herstein."
T.W. Hungerford, Algebra, Springer-Verlag. This is a graduate level textbook in the Graduate Texts in Mathematics series. It is a good reference.
A. Papantonopoulou, Algebra: pure & applied, Prentice Hall. This is a book with lots of examples. Its level is a little more sophisticated than our course, but you could still understand many of the examples.
J.J. Rotman, Advanced modern algebra, Prentice Hall. This is another graduate level textbook and could also serve as a good reference.

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