Math 113 - Section 4 - Introduction to Abstract Algebra

Instructor: Allen Knutson
Lectures: TuTh 9:30-11:00am, Room 70 Evans Hall
Course Control Number: 55086
Office: 1033 Evans Hall

The first midterm was approximately like this, the second like this. In fact, it was exactly this.
Office Hours: W 10:30am-12:00pm, 4-5 PM or by appointment
Prerequisites: Some exposure to linear algebra.
Text: Fraleigh, "A First Course in Abstract Algebra," 6th edition, Addison-Wesley
Grading: homework (30%), two midterms (20% and 15%) and a final exam (35%), 8-11 AM Friday Dec 15th.
Homework: weekly

Final stuff for this class...

There is lots of old homework sitting outside my door, and the latest homework has gone to the grader. Its solutions are lower down on this page.

There will be no practice final; I'm having writer's block. My apologies.

Rather, I will tell you what the topics are of the four questions:

  • nilpotents
  • classifying group actions on a set
  • invertible elements in a ring
  • Sylow subgroups
    • That's the course in a nutshell!

    Office hours are Tuesday, Wednesday, Thursday 3-5. Oops no: On Wednesday I can't make it 3-4, so it'll be 2-3, 4-5. Sorry for the late notice.

    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 (due Thursday 9/7). All page numbers refer to [Fraleigh]. Solutions (requires Acrobat reader)

  • p6 #16,17
  • p14 #6,9,10,11
  • p20 #3,6
  • (read p20 #7,8 too, but don't write them up)
  • Prove that "one-to-one" is the same as "monic", and "onto" is the same as "epic".
    • ("Monic" and "epic" aren't defined in the book; ask someone if you missed this definition. If you don't know the phone number or email address of anyone else in the class, you'd better swap with someone next chance you get!)

    Homework #2 (due Tuesday 9/19). All page numbers refer to [Fraleigh]. Solutions (requires Acrobat reader)

  • p16 #24-26
  • p38 #1-6
  • p61 #11-20
  • Find a graph whose automorphism group has exactly 3 elements.
  • Let G be a 10-vertex graph with a pentagon on the outside, a star on the inside, and each outer vertex connected to the corresponding inner vertex. Find automorphisms of G of orders 1,2,3,5,6.
  • Find examples of relations that are/aren't symmetric while they also are/aren't transitive while they also are/aren't reflexive, or else show they can't exist. There are 2*2*2=8 possible combinations to worry about.
  • If (G,*,inv,id) is a group, and (G,*,b,c) is also a group but now with b as the unary "inverse" operation and c as the identity, show that b = inv and c = id.
    • Homework #3 (due Thursday 9/28). Solutions (requires Acrobat reader)
  • p85 #26,32,33,40,45,52,54,56,62,63
    • Homework #4 (due Thursday 10/5). Solutions (requires Acrobat reader)
  • p102 #3-8,21,22,35,44
  • p189 #32,33,35
    • Homework #5 (due Thursday 10/12). Solutions (requires Acrobat reader)
  • Show that the Sam Loyd 14-15 puzzle is unsolvable, if exactly two pieces have been switched.
  • p115 #9,10,14-18,24,26
  • Let p be a prime number, and G = {1,2,3,...,p-1} under multiplication mod p. Show G is a group.
  • If p is prime, and n a non-multiple of p, show n^p (n to the power p) is congruent to n mod p. Hint: use the group G from above. Incidentally, this is a really quick way to show that some large number p is not prime without actually finding any factors!
  • p127 #29,30
    • Homework #6 (due Tuesday 10/31). Solutions (requires Acrobat reader)
  • p202 #5,6,8,9,10,13
    • Homework #7 (due Thursday 11/9). Solutions (requires Acrobat reader)
  • p204 #17
  • p222 #4,13
  • Find a 3-Sylow subgroup of S_9.
    • Homework #8 (due Thursday 12/7). Solutions (requires Acrobat reader)
  • p261 #14,15,19,41,46,47.
  • Show End(A) is a ring exactly if A is commutative.
    • Second practice midterm (with answers) much longer than the real thing