Math 417 C3M Home Page

This will be the home page for Math 417 C3M. "Introduction to Abstract Algebra". This class meets for the Spring 2019 semester at MWF 10:00-10:50 in 345 Altgeld. Office hours are (generally) M 4-4:30 in 241 AH and Tu 4-4:30 in 314 AH
The mathematical links page will be here. Suggest a new one!

Many students are concerned about the political issues of the day. If you want to contact your representatives, here are their numbers:
US Sen. Tammy Duckworth (D-IL) (202)-224-2859
US Sen. Richard J. Durbin (D-IL) (202)-224-2152
US Rep. Rodney Davis (R-IL 13) (202)-225-2371
IL Sen. Scott Bennett (D-IL 52) (217)-335-5252
IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

Test one is F 2/22 in the usual room at the usual time. Closed book, closed note, no calculator. One 3" x 5" card available; blanks available from class. Topics are summarized in Test 1-- Topics for Review 3. Office hours (not Monday at 4 because of special lecture): Tu 4 (341 AH) and Tu 7 (341 AH).

M 2/11, W 2/13, W 2/15, M 2/18 Catch-up -- No class 2/11 because of instructor's illness. We have reviewed permutations and proved LaGrange's Theorem. Lots of examples relating to subgroups and isomorphisms. Homeworks 2 and 3 were returned and discussed and the Homework 4 Solutions were distributed and discussed. Please let me know if I forgot anything.

F 2/8 -- Distributed Homework 4, due F 2/15 and the Homework 3 Solutions. The First Exam is tentatively set for 2/22, covering HW1-4, except as noted on HW4. In class, we found all the subgroups of S_3 (from last class' handout), talked more about isomorphism and subgroups of C_{18} and moved onto section 8 in the book. (We will skip section 7.)

Questions on HW3

  • Homework 3, due F 2/8.
  • Problem #3 is a repeat from HW2. Ignore it; it's a "free square."
  • A student asks about closure. My reply: A subgroup is closed under the operation, of the group which means that any product is in there. Consider H as a potential subgroup of G = (Z/6Z,+) and suppose you know that 1 is in H. Then 1+1 is in H, which means 2 is in H, so now you have to check 1+2 and 2+2, which have to be in H, so 3 and 4 are in H, and so 1+4 = 5 is in H, and in this way, you see that H has to be all of G.
  • In #4, yes, I meant that the generator is a, not x, but it really doesn't matter!

    W 2/6 -- HW1 returned. Main points include explaining your answers in words, not just with a number or two. I reviewed the definition of group, subgroup and cyclic group, proved that the subgroups of the cyclic group C_n = {a^i, i = 0,1,...,n-1}, a^n = 1, are generated by a^k, where k is a divisor of n. Changing gears, started talking about permutations (see section II.8), with particular emphasis on the order in which multiplication takes place. Distributed at the end an enlargement of the book's presentation of the symmetric group on three symbols and its multiplication table. You are encouraged to work out some of the products to make sure you see how it works.

    M 2/4 -- Distributed Homework 3, due F 2/8 and the Homework 2 Solutions. Gave the definitions for subgroups and cyclic groups and a bunch of examples. See Fraleigh I.5 and I.6. Talked a bit at the beginning about David Blackwell and the history of racism in Champaign-Urbana. Here are a couple of links: Column on segregation here lasting through the 1960s and Oral history of Prof. David Blackwell.

    Questions on HW2

  • 0. Owing to the cancelled class on 1/30, HW2 is now due M 2/4.
  • 1. Yes, another typo. In 5d, it should be the "usual additive group (Z/4Z)" without the star. (The additive part should help you remember.)
  • 2. In 3a, n = 168. In 3b, the question is to prove this for ANY integer n.

    F 2/1 -- Went over the worksheet mentioned in the 1/30 entry. Talked about isomorphisms and determined all groups of order 3 or 4. Also proved that if a is an element in a finite group then there exists a smallest positive integer n so that a^n = e. This is called the order of the group. For Monday: read sections 1.5 and 1.6 of Fraleigh, and look up one of your most illustrious predecessors as an Illinois math major: Prof. David Blackwell. (Added on 2/2: Here is the worksheet from class, both unfilled-out, and with the answers.)

    W 1/30 -- Stay home and stay warm. HW2 is now due on 2/4 (see above). Here is a link to the active learning questions I sent out in the email: work by yourself or with friends.

    M 1/28 -- Office hours will end at 4:25 today so I can catch a bus. Started talking in class about material from Fraleigh on binary operations and isomorphism and the definition of a group. I showed that (Z/nZ,+) and (Z/nZ*,x) are both groups, and worked out all the groups with 2 or 3 elements. There was a handout of number theory notes covering Friday's class. Here, as promised, is a link to a biography of Sun Zi, who is credited with discovering the Chinese Remainder Theorem. And here is a link to Francis Edward Su's guide to writing homework for math classes. I wasn't sure there was sufficient interest to xerox this and pass it out. Available on request.

    F 1/25 -- I passed out the Homework 1 Solutions at the beginning of the class and the Homework 2 Problems, due a week from today. Mathematically, I finished up with number theory, talking more about primes than before, and giving an alternate interpretation to the gcd, in terms of primes. There will be a handout on Monday or Wednesday. Please start reading the text, Fraleigh: we are headed for abstract algebra.

    Questions on the material and on HW1

  • I am happy to answer your questions and anonymize them on this webpage.
  • Explain your answers, don't just write them down. I will let you know if I think you should be writing longer (or shorter) answers.
  • As the mass mailing indicates, there is an error in HW1#4. I should be asking for 1/407, not 1/1407. The point of the problem is that you know something about the decimal expansion of 1/407, even if you don't do the long division.
  • A student asks about HW problem 4. I replied in part: "The key is to remember that if you write a/b = c/(10^n-1), then the fraction a/b will have the digit pattern of c repeated every n places. The example I gave in class was that 7/27 = 7*37/27*37 = 259/999, and as a fraction, 7/27 = .259259259... "
  • A student wanted clarification on some terminology. I replied in part: The expression "a | c" means that there is an integer n so that c = a*n. In words, either "a divides c" or "c is a multiple of a". Since c - 0 also equals a*n, a | c is equivalent to c = 0 mod a.
  • By "the power of 2" in #6, I mean: is it 2, 2^2, 2^3, etc.
  • In #3, find all x that satisfy both properties simultaneously.

    W 1/23 -- I passed out the phone/email list (not linked). Much of the time was spent talking about the handout on Repeating Decimals. Then moving on to the Chinese Remainder Theorem. Here is a Summary of what's going on, with the example I used in class. Outline maps of your hometowns were made available, as they will be for several class periods.

    F 1/18 -- I passed out an example of the Euclidean Algorithm, as done in class on 1/16. I also passed out the First Homework, which is due 1/25. I have gotten a few questions about it, and these will be pinned to the top until the homework is due. I will only answer questions about the homework until 1/24 @ noon. In terms of course content, I gave the general presentation of the Euclidean algorithm to compute the GCD and gave an important consequence: if gcd(a,b) = 1 and a | b*c, then a | c. I started also to talk about repeating decimals. There will be a handout on Wednesday and maybe even a short worksheet.

    W 1/16 -- Administrative material. The typical office hours will be Monday and Tuesday from 4-4:30. I will find room information, and announce it in class on Friday. The textbook is the 7th edition of Fraleigh. The link for the number theory material is Goodman's text . There may be a few homework problems there. The first homework will be due F 1/25. Mathematically, I gave an informal definition of a group, and I covered divisibility, a formal definition of Z/nZ and what "a = b mod n" means.

    M 1/14 -- Introductions. Handouts were: Course Organization, How to Solve It guide, Class questionnaire. and information about the links page (see above.) In terms of content, we gave some examples of groups involving rotations and flipping of squares and addition mod n and multiplication for restricted elements mod n.