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

Final scores: >200 - 1; 190s - 5; 180s - 6; 170s - 7, 160s - 4, 160> - 2. Course grades: A+ (1), A (8), A- (2), B+ (3), B (6), B- (4), C+ (1). You were an unusually enjoyable class to teach. Two notes on the final: even though only 3 problems could be dropped, a majority dropped each of 4 problems: 6,11,14,15. The most common error was in 13a (false, see #5). I don't post solutions, but if you want to know the answers in email, let me know. I'll be out of email contact after 6pm tonight for a while, but if you want to know your grade, send me an email with your booklet-number or test 3 score as a pin.

FINAL is F 5/3 @ 1:30 in the usual room.

• There will be no stereotyped "last two digits" or "repeating decimal" questions, but there will be something about Euler's Theorem, which leads us to it.

F 4/26 (Test 3), M 4/29 Test 3 back plus discussion, F 5/1 Final review, and Existence of primitive root handout, for "fun".

Final-type questions

• If anything about repeating decimals appears on the exam, it will be as part of one of the "choice" questions, or extra credit.
• To clarify about Ring homomorphisms: if R and S are both rings, then phi: R -> S is a ring homomorphism if phi(x + y) = phi(x) + phi(y) and phi(x*y) = phi(x)*phi(y) for all elements x, y in R, where the operations on the left of the equal sign are in R and on the right of the equal sign are in S. As a special case of these, if R = Z/3Z and S = Z/12Z and phi( [ 1 ] _3) = [a]_12, then using [1] + [1] + [1] = 0 and [1]*[1] = [1] in R = Z/3Z, you get restrictive information about a, and that is what I was doing today. With numbers other than 3 or 12, you’ll get a different answer in general.

Just to correct an unfortunate typo on the Course Organization, the Final in this class is Friday May 3, 1:30-4:30. (Not Monday, May 3). My plan for the week is to have your tests graded by tomorrow morning. I'll give some review and some reflections on teaching. Wednesday is the last regular class meeting. I'll do more review, and have the ICES forms.

Last minute comments on the test, which are not "obvious" from class discussion or the list of topics:

• This has come up a couple of times. I will not put repeating decimals on the third test. It might show up in one of the "choice" problems on the final, haven't decided yet. The basic idea is that if gcd(b,10) = 1, then 10^m = 1 mod b for some m dividing phi(b) and so a/b has a repeating decimal whose period divides phi(b). I'll go over this next Monday in class.

M 4/22, W 4/24 -- Lots of review. Here is the list of topics for the third test on F 4/26. Anything I've said about the existence of primitive roots (as opposed to playing around with them) will not be on the test.

W 4/17, F 4/19 -- Full web update will be done over the weekend, but here are the Homework 10 Solutions. (Actually, the wheels have fallen off my attempt to keep up, but all handouts from class are here.)

M 4/15 -- Here is the final version of Homework 10, the last homework of the semester. The third test will be F 4/26. HW 10 is due on F 4/19, and I will grade it myself over the weekend to get it back M 4/22. (Assuming I don't get hit by a meteor, etc.)

Here is Homework 10, subject to minor tweaks until it is distributed in class on Monday.

F 4/12 -- Homework 9 Solutions and a lengthier discussion about the material from 4/10 and the homework. The last homework, Homework 10, will appear over the weekend and be delivered in class on Monday. I need to figure out what about polynomials I can cover on 4/15 and 4/17. There will be new material after this, but it won't be on the exams.

W 4/10 -- T2 back and discussed. Then more about Fermat's Theorem and Euler's Theorem. Here is a summary of the reasoning behind gcd(a,100) = 1 ==> a^20 = 1 mod 100 (taken from an email query):

• gcd(a,100) = 1 <==> gcd(a,4) = 1 & gcd(a,25) = 1.
• gcd(a,4) = 1 ==> a^φ(4) = a^2 = 1 mod 4 by Euler.
• gcd(a,25) = 1 ==> a^φ(25) = a^20 = 1 mod 25 by Euler.
• So gcd(a,100) = 1 ==> gcd(a,4) = 1 & gcd(a,25) = 1 ==> a^2 = 1 mod 4 & a^20 = 1 mod 25 ==> a^20 = (a^2)^10 = 1^10 = 1 mod 4 & a^20 = 1 mod 25.
• a^20 = 1 mod 4 & a^20 = 1 mod 25 ==> a^20 = 1 mod 100, either by Chinese Remainder theorem (since gcd(4,25) = 1) or since 4 | a^20 -1 and 25 | a^20 -1 ==> 4*25 | a^20 -1.

M 4/8 -- A story first about beer, then about Section 20. Handouts eventually. Here is the (mostly accurate) Homework 9. Please note that the homomorphism in 2. is a RING homomorphism.

Here was Homework 9, at least up to typos and other corrections. The official version will show up in class on Monday.

F 4/5 -- Here are the Homework 8 Solutions. I solved the wrong problem and will correct this in class on Monday. The correct solutions will appear on the next set of solutions. In content, we have now covered sections 18 and 19 and will move to 20 on Monday.

W 4/3 -- Test. Nothing online

M 4/1 -- Mostly review for the test on W 4/3. The handout I gave was accurate, except that the date should have been 3/29/19, not 3/29/17. See you in office hours if you come, and I'll have homeworks and 3 x 5 cards for those who don't have them yet.

W 3/27, F 3/29 -- More discussion of rings and fields, and start of review for the second test. Here is Wednesday's worksheet. Here is the list of topics for the second test from Friday: list of topics

M 3/25 -- We started to talk about rings and fields; this can be found in Section 18 of Fraleigh. Homework 8 was distributed; see below.

Here is Homework 8, due F 4/5, after the second test. It will be distributed in class on 3/25. The traditional typo appears in problem 3a; that should be Z/10Z, not Z/0Z.

State of the class at Spring Break. I have covered all the material that will be on Test 2. After the break, I will start talking about rings and answering any lingering questions about groups. The test is scheduled for Wed. April 3, which is later than it "should" be, but will let you get back up to speed after the time off. Homework 8 will appear during next week and will be due Fri. April 5.

M 3/11, W 3/13, F 3/15 -- Here are the Homework 7 Solutions. Note that I got a sign wrong and 1a should be -19. Here is Wednesday's worksheet.

Bitten by the typo-bug again. In #3, should be 16, not 18. Here's the revision, to be distributed in class tomorrow. Homework 7.1

M 3/4, W 3/6, F 3/8 -- I've removed the typo-version of HW 7. See above. Here are the Homework 6 Solutions. We have now covered section 11 in the book. We're skipping section 12. We're in the middle of section 13. The second test will cover through parts of section 14 and 15 and won't happen until after Spring Break. There will be class on F 3/15, when Homework 7 is due, and I'll talk about those problems and other questions of interest. You should consider it sort of like an office hour. I know a lot of people will have already gone home.

• Revised HW 6, with problem 4 clarified thanks to alert student JR. Homework 6
• Here is Homework 6, but ignore it because it isn't the actual homework!
• A student asked about the various appearances of the dihedral group. This is what I wrote: Mathematicians are a little sloppy: when two groups are isomorphic, we tend to do something in one and assume without saying that it immediately applies for the other. To be precise, we have two isomorphic dihedral groups for all n \ge 3.
1. The group defined by x^n = y^2 = e and y x = x^{n-1} y.
The group of symmetries of a regular n-gon with vertices labeled {1,,,n}, where x corresponds to (12…n), the cycle on the vertices, and y (which I won’t write down) is the permutation corresponding to a flip on a vertical axis (though any axis will do).
3. In addition, and this is tricky, when n=3, there is a third isomorphism, to S_3, because every permutation of {1,2,3} can be viewed as a symmetry on a triangle.

F 3/1 -- Test 1 was returned and discussed. (Average = 86, Range = [71,97]). Versions of often-missed problems will appear on Homework 6, which will be available online by the afternoon of 3/1. The Homework 5 Solutions were distributed and discussed. We talked about the dihedral group D_4 in the context of its "abstract" formulation and saw what its subgroups and some cosets look like.

• As emailed, error in #4: the fourth element in D_4 should be (1432) not (1423).
• All the problems are do-able. Just look at the definitions. If you aren't sure how to prove something for a general group, see if you can prove it for special groups that you already know about. Note the patterns!

W 2/27 -- Summary review of cosets with examples and then a worksheet on S_3, as viewed as the dihedral group D_3. Class ended with a brief general discussion of D_n. We'll come back to this later. Edited to add: JZ identified a useful website with information on finite groups: Grouppropswiki

M 2/25 -- Review of cosets and overview of what's coming next in class; Homework 5 distributed. Note error fixed above

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. By "prime factorization" I just mean the Fundamental Theorem of Arithmetic, which says that every integer greater than 1 can be factored into a product of primes. Office hours (not Monday at 4 because of special lecture): Tu 4 (341 AH) and Tu 7 (341 AH).

• In response to email questions:
• The order of a group G is the number of elements in it. The order of an element x of a group G is the smallest positive integer k so that x^k = e. It is also the order of the subgroup of G which is generated by x: {e,x,x^2,...,x^{k-1}}.
• Tips for finding subgroups:≈ one way to start is to look at the subgroups which are generated by elements in the group. You then want to see if you can add more elements without getting the whole group. There is no easy algorithm to do "by hand".

Informal version of Homework 5, to be due on Friday 3/1. Hardcopies will be distributed in class on Monday, and it's possible that some typos will be fixed.

F 2/22 -- First exam. Won't be online.

W 2/20 -- Some review, including HW4 #4 done in complete detail. Lots of time spent on a worksheet about the Quaternion group Q_8, with one small typo: (-1)(-i) should equal i, not 1. The answer was that the only proper subgroups are those generated by the elements, and there is no subgroup which is isomorphic to V.

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.