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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

Th 12/7 -- Some additional comments for the test:

If you are doing division, say, in Z/11Z, then 4x = -7x and it
doesn't matter which one you use on an exam unless I say that all
ceofficients should be positive.

If you have a ring homomorphism phi from Z/2Z to Z/5Z and
phi([1]_2) = [a]_5, then [1]_2 + [1]_2 = [0]_2 and [1]_2*[1]_2 =
[1_]_2 imply that 2a = 0 mod 5 and a^2 = a mod 5, and so a = 0 and phi
is trivial.

If you have a ring homomorphism phi from Z/2Z to Z/6Z and
phi([1]_2) = [a]_6, then [1]_2 + [1]_2 = [0]_2 and [1]_2*[1]_2 =
[1_]_2 imply that 2a = 0 mod 6 and a^2 = a mod 6. In this case, there
are two possibilities: a = 0 and a = 3. Similar considerations apply
for ring homomorphisms from Z/nZ to Z/mZ.

A primitive root mod n, if it exists, is an element that is a generator for
(Z/nZ)^*; that is [a]_n so that if gcd(b,n) = 1, then there exists k with b =
a^k mod n. Such an a must have multiplicative order phi(n), so a^k is
not equal to 1 mod n if 1 \le k < phi(n). We have proved that
primitive roots exist when n is a prime.

A field is a ring in which multiplication is commutative, there exists a
multiplicative identity 1, there are no zero divisors, and every non-zero
element has a multiplicative inverse. What's been most relevant is that Z/nZ
is always a ring, but it is a field if and only if n is a prime.

W 12/6 -- Handout of an enhanced discussion of topics for test 3, to be held on F 12/8. A few other questions answered and then, the material of the handout was covered. The final topic was Hensel's "lifting" of primitive roots mod p to mod p^2.

M 12/4 -- Discussion of topics for Test 3 (on F 12/8). A written version (slightly revised) will appear on Wednesday. Other questions discussed and a long handout on Euler Phi, Inclusion and Exclusion and the existence of primitive roots was both passed out and discussed. More on Wednesday, after any review questions.

F 12/1 -- HW 8 returned. A somewhat out-of-order set of solutions to HW 9 was distributed. We talked about the Principle of Inclusion and Exclusion, applied it to the Euler Phi function, and were in the process of applying it to primitive roots when time ran out. Notes will follow next week.

HW 9 Questions.

Ok, a reminder about primitive roots. Phi(n) is the number of
integers a in {0,...,n} which are relatively prime to a, and
if n is prime, then phi(n) = n-1. For example, phi(6)=2: a = 1,5;
phi(7) = 6: a = 1,2,3,4,5,6; phi(8) = 4: a = 1,3,5,7. Phi(n) is also
the number of elements in (Z/nZ)*.

A primitive root mod n is an element a of (Z/nZ)* which generates
the whole group when you take powers. For example, if n=7, 2 is NOT a
primitive root, because the powers of 2 mod 7 are 1,2,4,8=1, and then
repeat. You are missing 3,5,6. But 3 IS a primitive root, because the
powers of 3 are 1 = 3^0, 3 = 3^1, 9 = 2 = 3^2, 27 = 6 = 3^3, 81 = 4 =
3^4,, 243 = 5 = 3^5 and 729 = 1 = 3^6. Each of {1,2,3,4,5,6} appears
in the list {1,3,2,6,4,5} in some order. The homework asks for
primitive roots mod 9 (phi(9) = 6) and 11 (phi(11)= 10)

W 11/29 -- More on polynomials and the content of section 23: factorization, the bound on the number of zeros and the Eisenstein criterion. A rather popular and involving worksheet

M 11/27 -- We return. I returned HW 8 and distributed HW 9, due on F 12/1. The content of the class was a review of rings and rings of polynomials.

F 11/24 -- Here is the online version of HW 9, due on F 12/1. I will have paper copies available on M 11/27.

M 11/20 -- As promised, and somewhat terse: solutions to the 11/15 worksheet. HW9 will come out later in the week, and, of course, paper copies will be available in class on 11/27.

F 11/17 -- Remarkably high turnout for the day before the break! I had HW7 graded and ready to return but completely forgot to do it. My apologies. I distributed the solutions to HW 8 and answered a few questions, mainly about what ring homomorphisms are. I also spent about 20 minutes on the romantic history of the quaternions (google it!) and used the opportunity to talk about the quaternion group Q_8, which has the property that it is not abelian, yet every subgroup is normal.

W 11/15 -- Second test returned, and gone over in some detail. And then more ring theory, including polynomials. Here is a worksheet, based on some of the more challenging exam problems, with solutions promised on 11/20.

M 11/13 -- Mostly number theory (handout to follow): on Fermat's Little Theorem, Euler's Theorem and the Euler Phi Function. I also distributed a copy of notes about the ring homomorphisms from Z/2Z from 11/10. HW 8 is due on Friday, and there will be class. I will have to cancel W's office hours because I am on a math department grad student panel talking about how to teach diverse student populations. Let me know if this is a problem.

F 11/10 -- I distributed the solutions and there were no questions, oddly. I suspect there will be more on Monday. Most of the discussion was about rings, along with a detailed discussion of ring homomorphisms from Z/2Z, and there was a worksheet. Finally, HW 8, due F 11/17, was distributed. There will be class every day next week, but no office hours planned on W, because I'm on a grad student panel. We'll talk about this in class on M 11/13.

HW 7 Questions (temporarily pinned to the top).

Notes on HW7. On #5,6, just to be clear, 0_R = (0,0) and 1_R =
(1,1).

A student writes: I don't understand what you are asking for in
#4 when you say, "compute (x+3)^2 in R[x]" I know that (x+3)^2 = X^2 +
6x + 1 in mod 8. Would you clarify how I can "compute" further?

My reply: I don't see a lack of clarity in your answer. You might find
something interesting when you compute (x+7)^2

A student writes: I am a little confused on
the meaning of "unity of characteristic 3" in #2.

My reply: It's two things: ring with unity means that 1_R exists
in the ring.
Characteristic 3 means that for any element x, x+x+x = 3x = 0. So, (a + b)^3
= a^3 + 3 a^2 b + 3 a b^2 + b^3 as usual, but 3 a^2 b = 0 and 3 a b^2 = 0,
so, actually, (a + b)^3 = a^3 + b^3 in this ring.

A student writes: For (3)b: Determine the units in R= z/14z:
I found the units to be 3,5,9,11 because [redacted] am I on
the right track?

My reply: you are on the right track, and you are missing the two
easiest-to-find units.

W 11/8 -- Test 2, not online.

Notes on Test 2, which by class vote will be in class on W 11/8.
Will cover HW's 4, 5, 6.

1. A clarification on normal subgroups: A subgroup H is a normal
subgroup of G if aH = Ha for
every a in G; that is, every left coset is a right coset. I mentioned
3 cases in which you know automatically that H is a normal subgroup:
1. If G is abelian, then every subgroup is normal.
2. If phi: G -> H is a homomorphism, then ker(phi) = {g in G : phi(g) = e_H}
is a normal subgroup of G
3. If |H| = (1/2) |G|, then H is a normal subgroup of G.
This can happen in other cases, but these are three in which you don't have
to PROVE it's normal, but you have a theorem so it's automatic.

Notes on HW7. On #5,6, just to be clear, 0_R = (0,0) and 1_R =
(1,1).

M 11/6 -- Mostly review for the test. Distributed a copy of an example discussed on 11/3. Some ring theory, explaining any topics on HW 7 and the field Q(square root of 2).

F 11/3 -- Distributed the list of topics that are good for review for the second test. More on rings and polynomial rings. Also, distributed HW 7, due F 11/10.

W 11/1 -- Finished off the group theory part with another worksheet and started rings!

M 10/30 -- HW5 back, answering questions and doing more examples. Thm. 14-11 in a great deal of detail, to be finished on W 11/1.

F 10/27 -- Distributed the solutions to HW 6, and went over the material in detail. More review on questions in sections 14 and 15 of the book. We had a worksheet too.

HW 6 Questions (temporarily pinned to the top), along with other
relevant questions from your emails.

(Rest of the comments added 10/26)

On question 4, G is not cyclic, but it is the direct product of two groups that are cyclic, so a typical element is (a^j, b^k), where a^16 = 1 and b^42 = 1. Guessing for a correct answer is a good starting strategy, but in the end, you have to prove that the element has the order you need.

In question 6, please remember that aH = Ha just means that the two sets are equal. It does not mean that ah=ha for each individual element h in H. (Remember the example from S_3).

Answer for worksheet 10/20 (iv): You need to find j and k so that the LCM of the order of a^j and the order of b^k is 9, and the easiest way to get order 9 in the group generated by b. (Hilarious fact: if I use the notation we usually use to describe the group generated by b, it makes the rest of the page in boldface!) is to take b^2. For a^j, you then need that a^{9j} = e, which means that 12 divides 9j, or 4 divides 3j, so 4 divides j. so a^0 = e, a^4 and a^8 would work. Please note that the numbers are different from those on the homework!

In the book's theorem 11.12, the factors of Z come into play when G is a group with infinitely many elements. We're shown that G x H is isomorphic to H x G, and this isomorphism generalizes to any number of factors.

W 10/25 -- Distributed classnotes covering M's class and also started to talk about normal subgroups, factor groups and the connection to homomorphisms. We will schedule the test on F 10/27.

M 10/23 -- A long and careful discussion of gcd, lcm, and the orders of elements in C_n and C_n x C_m, as well as potential homomorphisms between two cyclic groups. Handout on W 10/25.

F 10/20 -- HW4 returned. Rather than a retro, I went through the questions individually. HW5 collected; the solutions were discussed. Also, HW 6 was distributed. We talked about cyclic groups, abelian groups, and the orders of elements in the product of cyclic groups. The lcm plays a role. We had a worksheet on the topic, and there will be a handout on Monday. For some reason, I forgot to post the HW 4 solutions last week, so here they are. I'm also adding them to F 10/13, because that's where you'd look for them.

HW 5 questions

Here is
HW 5.
Apologies for the delay for those who like doing homework on Friday
and Saturday nights.

On #2b: Look for an element of order 10. If every element of your
group is a power of this element, then what you have is a cyclic group
of order 10.

<
Sorry #4 looks like a permutation. I should have written it as two
separate triples. So, what I'm looking for is more like

( \Phi_1([0]_3], \Phi_1([1]_3], (\Phi_1([2]_3] )

( \Phi_2([0]_3], \Phi_2([1]_3], (\Phi_2([2]_3] )

"Showing by tables" means here that you show you have an isomorphism by
demonstrating that the multiplication tables are the same (up to renaming
the elements by the isomorphism.

In #6,
Every element in H can be written as b^k for some k (e_H = b^0 for
example). The most interesting element in G is its generator, so you might
want to look at phi(a), which would have the form b^k. You want to derive a
contradiction from phi(a) not equal to e_H.

W 10/18 -- HW4 should be graded by F, which is when HW5 is due and when HW6 should be distributed. Some more on direct products and a handout describing the isomorphism between G x H and H x G in excruciating detail. Also, lots more on homomorphisms, which are like isomorphisms but don't have to be 1-1 and onto.

M 10/16 -- Attendance is beginning to drop a bit. Don't go slack; more than any other subject, math classes are hard to catch up on! Content: some more review of cosets, direct products (both abstractly and in very concrete form, and an introduction to homomorphisms. Lot's more on that. Also: worksheet on C_4 x C_2. And the paper copies of HW 5 were distributed; this is due F 10/20.

F 10/13 -- First exam returned and HW4 discussed. Introduction to direct product of groups. HW5 delayed until the weekend; will be distributed in class on M 10/16. Added 10/22: HW 4 solutions.

HW 4 Questions

Typo alert: In Problem 4/5 of HW4, there is a typo. The 4th
element of the group should be (1432) not (1423).
(This was sent to everyone in the class (some twice) on Wednesday
night. If you did not receive the correction, please let me know.)

W 10/11 -- Test not graded yet. Lots of stuff on cosets, both in principle and in practice. I also spoke about the 14/15 sliding tile problem. The Wikipedia article has a lot more information, including history and generalizations.

M 10/9 -- First exam. Not posted online.

Test Questions

The first exam, by class vote, will be on M 10/9 in class.

The material is essentially that of the first three homeworks.

There will be a list of topics distributed later in the week.

Here is the link to the
list of
topics.

(added 10/8) I meant to write "associativity" on the sheet, and
I meant to say "isomorphism" instead of "automorphism".

As mentioned in class today, the Chinese Remainder theorem will
NOT be covered in the first test.

Test will be closed book / closed notes, except for the 3 x 5
card.

By request: a summary of why
isomorphisms
preserve inverses.

(added 10/8) And now, a summary of answers to various e-mail questions:

The lines in Fig. 6-18 in the book refer to inclusion of
subgroups, so <9> is a subgroup of <3>, <6> is a subgroup of <2> and
<3>, etc.

The Klein Group V is the group of order 4 which is not cyclic. See
Table 5.11

The identity element for a subgroup H of G will always be the same
as the identity element for G.

A binary operation is a map from pairs in a set S to the set
S. Closure is a part of the definition (that the range is in S). This
term comes up when checking subgroups, because if H is a subgroup of G
and h, h' are in H, then h*h' must also be in H. (But, this isn't the
only condition.)

The only congruence we've talked about is the usual "mod n" for
integers.

I meant to write "associativity" on the sheet, and I meant to say
"isomorphism" instead of "automorphism".

F 10/6 -- Various review questions, mainly about isomorphisms and how you know a subset of a group is a subgroup. I talked on some things that won't be on the test, like a listing of the subgroups of D_4. last topic was an introduction to left cosets and right cosets, which will show up on HW 4,

W 10/4 -- Homework 3 returned, with comments and retrospective, and a related worksheet. At the end of class, another worksheet on D_4. Lots of time answering questions and happy to spend more on Friday!

M 10/2 -- More on permutations and S_3 and D_4 and even and odd permutations. Much time was spent on a worksheet on D_4. We'll return to it on F 10/6. Also, HW 4 was distributed, due on F 10/13, which is after the first test. The material on this homework will be covered on the second test, not the first.

F 9/29 -- The third homework solutions were distributed, and discussed in detail. New material was mainly a discussion of permutations and the cycle notation. Homework 4 will be distributed on Monday, but will not be due until after the test.

HW 3 Questions

A student asks about what's going on in two of the Fraleigh problems:

In 1b, you consider the set {hk} of all elements in G where h
is in H and k is in K. You should check that this set satisfies the criteria
of Theorem 5.14.

In problem 2b, the element is a lower
case "sigma". So, sigma is in S_6 and is a permutation on the set S =
{1,2,3,4,5,6}. That is sigma: S -> S. You can calculate sigma^2 by applying
the function twice.

W 9/27. Homework 2 was returned, along with a retrospective, which led to a short worksheet. In terms of new material, we completed the discussion of groups of order six. On to cycle decompositions (section 9) on Friday.

M 9/25 -- No handouts. We completed section 8 with an informal discussion of dihedral groups (the symmetries of a regular n-gon under rotation and flips) and Cayley's Theorem. Then a jump ahead to Lagrange's Theorem (Thm. 10.10) and a discussion of all groups of order 5, 6 and 7. Groups of prime order are easy to describe. The ones of order 6 are more complicated and there will be a handout on W 9/27, along with the return of HW2.

F 9/22 -- Another Friday. Second homework solutions distributed, and briefly discussed. Also HW 3 distributed, due on 9/29. We finished up the discussion of section 6 on subgroups of cyclic groups and moved on to section 8 and groups of permutations. The end of the class had a of the handout from M 9/18, reprise of the handout from M 9/18, with instructions to identify this group as the S_3; that is, the group of 6 permutations of the set {1,2,3}.

HW 2 Questions

The book page numbers for HW2 problems are correct to within 1 or 2.

Typo on #6: I want the subgroups of the additive group Z/12Z, so
just delete the star.

To prove that a map between two groups is an isomorphism, you
need to show that it is bijection on the elements of the sets, and
that the operation of the groups is preserved. This can most easily be
done by giving the tables.

W 9/20 -- Homework 2 was returned and a retrospective was offered. Lots of discussion at the beginning of the class on subgroups and cyclic groups and we continued by describing all the subgroups of a cyclic group. We'll finish a theorem on subgroups generated by elements of a cyclic group. Our friend, the gcd, will return.

M 9/18 -- Graded homework should be returned on W. I reviewed the groups of order 4 and distributed notes on the derivation of the two such groups (up to isomorphism). There was also a worksheet which gave you a chance to work out the ideas on the group table for a group of order 6. (Spoiler: it's the symmetric group S_3.) The new mathematics covered was the idea of the subgroup, with some examples, and at the very end, the cyclic group

C_n: = {e, x, x^2, ... x^(n-1)}, x^n = e, where x^2 = x * x, x^3 = x^2 * x, etc.

F 9/15 -- First homework
solutions
distributed, and briefly discussed. Also
HW 2
distributed, due on 9/22. In class, we talked a bit about the
solutions, about the idea of group isomorphism and group tables. We
went through all the groups of order 1, 2, 3 and 4 (up to
isomorphism). There will be a handout on 9/18 and a worksheet to let
you play with this as well. The next new topic is the very important
one of subgroups, which will soon be followed by cyclic groups.
Read I-5,6 in *Fraleigh*.

HW 1 Questions.

A student writes: I have a question on #2 in the HW. It says as a
hint try to find several
triples of numbers which satisfy the conditions, but I can only come up with
one triple which would satisfy this: a=12, b=4, c=6, so g=2.

My reply: Actually, there are lots of examples. Consider a = 12*5, b =
4*7, c= 6*7, for example. But the primes 2 and 3 are special. Since gcd(a,c)
= 6, you know that 3 divides both a and c. If 3 divided b, then 3 would
divide gcd(a,b), but it doesn't. I'll talk about this in class on
Wednesday.

A student asks for advice on #3a.

My reply: Look at the power of p that divides n; that is,
v_p(n). If you write n^2 = n*n, keep in mind that gcd(n,n) = n and not
1.

A student asks for advice on #3b.

My reply: Well, without giving away the store here: think about
what information this
problem is telling you about the power of 2 that divides a, b, c. Does this
information tell you how to find a, b, c or how to show that no such a, b
and c can exist?

A student asks for advice on #6.

My reply: we know that a/b has a period of n if 10^n = 1 mod
b. Both 37 and 41
are prime, so we want both 37 and 41 to divide 10^n - 1 for the right choice
of n. I can't say much more without giving away the answer.

W 9/13 -- More discussion on gcd and v_p(n). A worksheet. Beginning of the definition of groups.

M 9/11 -- Two handouts. The first was notes for 9/8 and 9/11. The second was a worksheet which was discussed in class. I will start talking about groups on W 9/3; glance through pp.11-43 in Fraleigh. I won't be covering everything there. Edited to add: Here is a link to a biography of Sun Zi, who is credited with discovering the Chinese Remainder Theorem.

F 9/8 -- Three handouts. One was the phone/email list, which won't be posted for security reasons. Another consisted of notes on v_p(n), which you can find at W 9/6. The third was HW1. This is due on F 9/15. Send me questions if you have them, but the last few problems involve stuff we will cover on M 9/11. The main results will be on a handout Monday.

W 9/6 -- Went over factorizations and began congruence mathematics. Here is a link to notes on factorization, with a new and useful notation, v_p(n). I'll talk about this in class on Friday. We also did a worksheet.

M 9/4 -- Labor Day! Here is a history.

F 9/1 -- Optional placement on the map. The outlines will be
available every day next week. Mostly going through the Goodman
notes. We've finished with 1.6, except for unique factorization into
primes and how it influences gcd. There will be a worksheet on the
Euclidean algorithm and on congruences for W 9/6. No scan of
notes. The only new math not there is this sequence of definitions. If
S and T are sets, so that S = {s : s is in S} and m and b are numbers, then

We will be using this mainly when S = Z is the set of integers or S = mZ. The big theorem on Friday was that mZ + nZ = gZ, where g = gcd(m,n).

W 8/30 -- Discussion of the worksheet. Informal discussion of what the
definition of a group is. Here is a scan of a
Summary
of the parts of the day's lecture not in *Goodman*. Number
theory reference is
Goodman's
text . Please let me know if this doesn't work. I covered
roughly pp. 24-30, and hope to do much of sections 1.6 and 1.7.
The pronunciation of your names seems to be converging on what you'd like.

T 8/29 -- Based on questionnaire responses, and very tentatively, there will be office hours W 4->4:30 and Th 5-> 5:30 in 145 Altgeld. In each case, the rule will be that if nobody is there for 5 minutes, I leave, so come early!

M 8/28 (post-class) -- First question! .
A student writes: I was wondering if you can clarify a question on the
work sheet? Numbers 1 and 2 asked us to identify "which group of order
4 do they look like". I'm not exactly sure how to answer this. Are we
supposed to draw something here?

My reply: Good catch. I didn't get as far as I wanted in class. In the
first group, there was one element whose powers determined everything. It
was like {e,x,x^2,x^3}. In the second, it's {e,x,y,z} where x^2=y^2=z^2=1.
You don't have to turn these in anyway, so just get as far as you can.

M 8/28 -- First day of class. We had an informal introduction to the
material of the semester.

Here is a scan of a
Summary
of the first day's lecture, and a scan of an intended
Worksheet
that instead was used as an exercise in advance of the W 8/30 class.

Handouts:
Course
Organization, How to Solve It
guide, Class questionnaire,
emergency
guide from the UIPD.