Math 113 - Section 003 - Introduction to Abstract Algebra

Instructor: Allen Knutson
Lectures: TuTh 8:10-9:30am, Room 71 Evans Hall
Course Control Number: 54870
Prerequisites: Some exposure to linear algebra would be nice.
Text: Dummit & Foote, "Abstract Algebra," 3rd edition, Wiley.
We're not going to cover the whole book, but (a) it's impossible to find one that includes only the stuff I want, and (b) this one will continue to serve you well beyond this class.

Office: 1033 Evans Hall

Office Hours finals week: I should be in my office 11-3 every day (then at tea in 1015). You should check out this, too; in particular it lists the topics to expect on the final.

GSI: Tamás Kálmán, who last I heard has office hours Wednesday 9-11 and 1-4, Thursday 12-3:30.
Grading:

  • homework (30%),
  • midterm #1, Feb 17th, 15%. Topics to expect:
  • Proofs, especially induction
  • Partitions and equivalence relations
  • Automorphism groups of graphs
  • Permutations
  • Groups and group homomorphisms
    • Open notes, closed textbook. You don't need a blue book.
  • midterm #2, April 8th, 20% -- open notes closed text again
  • final exam (35%), 8-11 AM Saturday May 22nd, in 70 Evans (not 71!).
  • Final is open notes, closed book, like the midterms
    • Nota bene: I design my tests to not be really time-consuming, nor too dependent on tricks... but nonetheless to be really hard, usually with a median around 50%. If you think this is really unusual, I suppose you're right, but if you think it's really a bad idea, ask me about it. Calendar: Here's what we've done so far, and here's where we plan to go next.
    Homework: weekly, due Thursdays
    I expect to cover
  • Chapters 1,2,3,4,7,8,13, deeply
  • Chapters 5,10,12,14,15 & Appendix ][, barely
    • Probably you should read through Chapter 0 right now.

    If while looking through a chapter/appendix on that second list, you find something you definitely want to hear more about, let me know, and I may be able to find space. Or maybe you'll just hear about it during an office hour.

    I've been asked for some study aids. Here's what I've come up with.

    If you haven't done so yet, please print and fill out this survey and bring to class. (I know, it should be a web form like this, but I haven't learned how to do that and I'd rather have paper anyway.)

    Set theory

    If you're curious about the set-theoretic underpinnings of this class (and mathematics in general), check this out. You'll need Adobe Acrobat Reader (which you almost certainly have; if not you can get it here).

    Homework

    In general homework will be assigned Thursday and due at the beginning of class the next Thursday. I do plan to go over homework, but not all the problems; ask me Tuesday which problems you'd like discussed on Thursday. (Yes, this is to cajole you to look at the homework before the last minute!)

    Obviously you can come ask me questions at office hours on Wednesday -- if I think you should think more about the problem, I'll just say so.

  • Homework #1 has pictures, so I put it on its own page. Answers here
  • Homework #2: Answers here
  • Find a graph with exactly three automorphisms (not six).
  • [DF] p21 #1,5,6,9,12,15,27
  • Homework #3: [DF] Answers here
  • 1.2 #2,3
  • 1.3 #2,3,6,16
  • 1.6 #4,6,12,17
  • When a problem says "show these are isomorphic", you should write down an isomorphism.
  • Homework #4, due Feb 26th: Answers here
  • 1.7 #8,15,16,17,23
  • 2.1 #2,5,6,9,10,11
  • Homework #5, due March 5th (oops, should have been 4th, but whatever): Answers here
  • Determine the conjugacy classes in the group D_{2n}. Careful: you will probably want to separate into the cases n even and n odd.
  • 2.2 #3
  • 4.1 #1,2,3,4
  • Homework #6, due March 11th: Answers here
  • Let G = Z_2 x Z_2 x Z_2, with elements 000,001,010,100,101,011,110,111 added mod 2. Find three automorphisms of G, of order 2,3, and 7. (This is related to the Fano hypergraph I drew at the end of class; can you see how?) In fact |Aut(G)|=168 and its only prime factors are 2,3,7.
  • 4.1 #9a, 10
  • Let G = the rotations of the cube, so |G|=24. Let V = the 8 vertices of the cube. Let VxV = the set of ordered pairs of elements.
  • How many orbits are there of G on VxV? For each orbit, say how many elements of VxV are in it, write down one of them, and describe the stabilizer.
  • Answer the same questions for P = the set of 2-element subsets of V (note that these are unordered: |P| = 28).
  • Find a normal subgroup K of S_4, and a normal subgroup H of K, such that H is not a normal subgroup of S_4. ("Normality isn't transitive.")
  • Homework #7, due March 18th: Answers here
  • 1. If A,B are normal in G and have intersection {1}, show they commute.
  • 2. If H is a subgroup of G, what is the kernel of the action of G on G/H?
  • 4.3 #5,8,10,13,35
  • Homework #8, due April 1st (really!): Answers here
  • Let G be a group of size n, p a prime, such that |Syl_p(G)| = m > 1. If n doesn't divide m!, show that G has a nontrivial normal subgroup (neither {1} nor G). [A harder question, that I'm not asking: show the same for n not dividing m!/2. This is relevant for n=112.]
  • Let p^1 || |G|, and |Syl_p(G)| = m. Show that there are exactly m(p-1) elements of order p in G.
  • Let G be a group of order p^k, so Sylow's theorems become useless. If k>1, show that G has a nontrivial normal subgroup, i.e. isn't "simple".
  • Let G be a group of size n, n at most 100. Show that unless n is prime or 60, that G is not a simple group. If you can't handle all the cases, give a list of the cases that you get stuck on.

    • Here are examples, for n=12, of the sort of reasoning I expect.

    "There are either 1 or 3 Sylow 2-groups. If there's just 1, it's normal. If there are 3....

  • ...then since 12 doesn't divide 3!, the first HW question guarantees us a nontrivial normal subgroup." or
  • "...then there can't also be 4 Sylow 3-groups, as follows. The second question gives us 4*2 elements of order 3. That and the identity account for all but 3 elements, which would only enough room for a single Sylow 2-group. Hence, there is only 1 Sylow 3-group, so it's normal."
  • Homework #9, due April 15th: Answers here
  • 5.4 #10,13,15
  • 5.5 #2,6,9 oops #9 was a mistake
  • Homework #10, due April 22th: Answers here
  • 2.3 #1,12
  • 7.1 #2,3,11
  • 7.3 #7,8,10
  • Homework #11, due April 29th: Answers here
  • 7.1 #8,13,15
  • 7.4 #7,8,13,15,19,30
  • Homework #12, due May 11th (final class): Answers here
  • Let R = Z[t]/(t^2). Show that R is isomorphic to a subring of the ring of 2x2 upper triangular integer matrices. Find all the ideals of R.
  • Let M be a module over a commutative ring R, and r an element of R. Let rM denote the subset of M consisting of all {rm : m in M}. (One might call this a "principal submodule".)
  • If r is a unit, show that rM = M.
  • Give an example where rM is smaller than M.
  • Give an example where r is not a unit, M is not zero, but rM = M nonetheless.
  • Let M be a left module over R, and m an element of M. Let I be {r in R : rm = 0}. Show that I is a right ideal, the "annihilator".
  • 13.1 #4
  • 13.2 #2,4,14,19

    • Here are the study aids I suggested over break. Here's some more for the final.