Math 109, Fall 2012

Akshay Venkatesh, MWF 9--9:50 in 380F.

• The Platonic solids assignment: When proving two groups A and B are isomorphic, your proof should clearly and visibly address each of the following steps:
  • Explicitly define a function F from A to B.
  • Explain why F is a homomorphism.
  • Explain why F is 1-1.
  • Explain why F is onto.
  These should not be done "implicitly". Each of these steps should be spelled out and visible.
• The elliptic curves assignment: These needed more details on basic points, related to precisely and accurately defining the group elements and the group law. For example: What is the point at infinity, and what is its role? When one adds two points, what happens if the line through them is vertical?

Here is a practice final that you can work on. This is a REVISED VERSION as of Friday March 16 11pm -- the old version was a bit too hard. Here are some solutions. Sorry, they were very hastily written, and a couple of the more routine problems are omitted.

I will have extra office hours Friday 3--4 and Jenya will have extra office hours Friday 4--5.

Text: "Groups and Symmetry" by M.A. Armstrong.

Office hours: My office hours will be MWF after class (10 - just before 11), in 383-E. The course CA is Jenya Sapir. Her office is 381-H, on the first floor. Jenya's office hours will be W 4-6, T 1:30-3:30 (note! change from M to W).

The midterm has been graded (out of 50). The median score was in the high 20s. You can see your grade on Coursework. Were your grade based on this score alone, the A/B cutoff would be in the low 30s, and the lower cutoff for Bs would be around 20. If your score was below 15, please come and see me or the CA.

Solutions.

Some common errors: On 1(b), the order is the SMALLEST power of an element which equals the identity. For problem 3(b), to show that C need not be a subgroup, you need to give a specific example of A, B and G such that C is not a subgroup. It is not enough simply to say that the proof for 3(a) breaks down when G is abelian -- how do you know there isn't some other proof which works? Problem 5(b): A number of people claimed that the negative of the sign homomorphism is also a homomorphism. It is not: Any homomorphism sends the identity to the identity.

WIM assignment info: The WIM due-date will be Friday, March 9. A draft (which will not count towards your grade, but you are strongly encouraged to hand it in) will be due on Wednesday, February 29. Details.

• HW 1, posted Friday Jan 13, due Friday Jan 20: 2.3, 2.6, 2.7, 2.8, 3.2, 3.3, 3.5, 3.7, 3.8, 4.7.

  Bonus question: Prove that if R, S are two rotations of the sphere, then "R followed by S" and "S followed by R" are rotations by the same angle, although possibly through different axes.

  Solutions by Jenya.

• HW 2, posted Sunday January 22, due Monday January 30: 4.4, 4.5, 4.8 (note how this is related to the bonus problem from HW1). 5.1, 5.4, 5.5, 5.7, 5.11.

  Solutions by Jenya.

  Common errors: In 4.5, you should make sure to remember that the order is the SMALLEST power equal to the identity, not just any power equal to the identity! In many cases, for problems 5.1, more justification was needed on why your list of subgroups was complete. Also the group itself is a subgroup! In 5.5, the criterion for being a subgroup was often misapplied. (If you aren't sure what you did wrong, please talk to me or the CA.)

• HW 3, posted Sunday January 29, due Monday January 6: 6.1 -- 6.9 (nine exercises), 6.11, 7.1. For 6.4, there is no need to prove your answer about the number of subgroups.

  Bonus question: Explain why that one cannot solve a Rubik's cubes using only legal cube moves after one flips an edge piece.

  Solutions by Jenya.

• HW 4, posted Sunday February 5, due Monday February 13: 7.4, 7.5, 7.6, 7.7, 7.9, 7.12.

  Bonus question: Let G be a group, with multiplication law *. Prove that the rule x#y = y*x also defines a group law on the elements of G (i.e.: reverse the order of multiplication). Prove that this new group is isomorphic to G.

  Solutions by Jenya

• HW 5, posted Monday February 13, due Tuesday February 21: 11.2, 11.3, 11.5, 11.7. Compute the order of 2 in the multiplicative group modulo 59. Find an integer x such that x^71 is congruent to 3 modulo 1001.

  No bonus question this week. Have a look at the WIM assignment instead! Solutions by Jenya.

• HW 6, posted Wednesday February 22, due Wednesday February 29: 9.1, 9.4 (only the equilateral triangle part), 10.1, 10.2, 10.3, 10.4, 10.6, 10.12.

  Bonus question: Explain why, for R an element of SO(3), the rotation angle of R is determined entirely by the trace of R (i.e., the sum of the diagonal entries). Now solve the bonus question from HW 1 using this fact.

  Solutions by Jenya.

• HW7, posted Friday March 2, due Friday March 9: 14.1, 14.4, 14.5, 15.2 (enough to do D_4, D_5), 15.4, 15.6. Bonus question: 15.11.

  Solutions.

• There is no homework 8. However, here are some problems which it may be helpful to look at in relation to the material of the last couple of weeks: 16.1, 16.2, 16.8, 17.1, 17.2, 17.3, 17.4, 17.5, 17.10. Here are Solutions.

• Week 1: Rotations of a cube. Definition of a group. Examples: integers, real numbers under addition. Integers modulo n. Dihedral groups. (Chapters 1--3 of text).
• Week 2: More on dihedral groups. Order of a group and order of an element. If a group has finite order, every element in the group has finite order. The notion of subgroup, the subgroup generated by an element. (Chapters 4--5 of text.)
• Week 3: The group generated by a set of elements: it is the intersection of all subgroups containing that set. Permutation groups. Shorthand for permutations. Every permutation is a product of transpositions. The sign of a permutation: Even permutations form a subgroup of half the size. (Chapters 5--6 of text)
• Week 4: Proof that the notion of even/odd permutation is well-defined. Homomorphisms. The symmetry group of a cube is isomorphic to S_4, by considering how it permutes vertices. (Chapter 7 of text)
• Week 5: Every group is isomorphic to a subgroup of a symmetric group, with proof. We skipped Chapters 9 and 10 and started on Chapter 11: Lagrange's theorem. Proof of Lagrange's theorem. Consequence: order of an element divides the order of a group. Fermat's little theorem.
• Week 6: The midterm. RSA and Diffie-Hellman. Matrix groups. (Chapter 9 of text; RSA and DH are not covered in the text.)
• Week 7: Products (Chapter 10 of text). An abelian group of order pq, with p,q relatively prime, is isomorphic to the product of the subgroups with g^p=e and g^q=e. Conjugacy classes (Chapter 13 of text, referring to Chapter 12 as necessary).
• Week 8: Conjugacy classes in the symmetric group, in the dihedral group, and in A_5. Normal subgroups are kernels of homomorphisms. If f: G--> H is a homomorphism, then the order of G is the order of the kernel of f multiplied by the order of the image of f. A_5 has no normal subgroups. Any homomorphism from S_5 to a smaller group is determined by the sign. This covered parts of Chapters 14, 15, 16. You don't need to know material from the book that we did not cover, e.g. quotients.
• Week 9: A_n has no normal subgroups. The homomorphism from S_4 to S_3 and its kernel. Orbit-stabilizer theorem, first version.