This course satisfies the Writing in Major requirement. In addition to learning the material, we shall focus on writing clear and complete mathematical proofs. In your homework assignments, you will therefore be required to write in complete, grammatical sentences; you will be graded both with respect to mathematical accuracy and for clarity of expression. If you find writing proofs difficult, please do talk to Josh or me during office hours.
About two thirds of the way through the course, there will be a writing assignment due. I will give details of this project later. The writing project is an integral part of the course, and it applies also to non-math majors.
Prove that the average number of fixed points equals the number of orbits.
An element g of G is called a derangement if it has no fixed points. Prove that if the group action is transitive, and S has more than 1 element, then there must be element of G which is a derangement.
Consider the symmetric group S_n acting in the usual way on the numbers from 1 to n. Can you find a formula for the number of derangements? As n goes to infinity, show that about n!/e elements are derangements (e=2.718...). What happens when S_n acts on subsets of k elements from 1 to n?
In writing this up, you should include an introduction explaining what a group action is, what it means to be transitive, etc. Then you should try and explain as clearly as you can the proofs of these results. You should also feel free to add any supplementary results about group actions that you think may be illuminating, or work out examples to illustrate your results. If you have any difficulties with figuring out the math or with writing please get in touch with Bob Hough (who is our WIM grader, 380G) or me. Bob will hold office hours next week (May 18-22) on Tuesday and Thursday from 4-6, and Wednesday from 2-4. You should have a first draft of your project done by May 22. You'll have comments back from Bob by May 29, and then the final draft will be due back on June 1.
Week 1: Due Friday, April 10 (in class). Section 1.1: problems 1, 5, 7, 18. Section 1.2: problems 3, 7. Section 1.3: problems 15, 18. Section 1.4: 2, 6, 8. Section 1.5: problem 1. Solutions by Josh Genauer.
Week 2: Due Friday, April 17 (in class). Section 1.6: 2, 6, 7, 17, 26. Section 1.7: 8, 16, 18. Section 2.1: 2, 4, 9, 12. Solutions by Josh Genauer.
Week 3: Due Friday, April 24 (in class). Section 2.2: 1, 5, 6. Section 2.3: 10, 11, 16, 24. Section 2.4: 13, 15. Section 2.5: 2, 9. Section 3.2: 6, 8, 11, 22. Solutions by Josh Genauer.
Week 4: Due Friday, May 1 (in class). Section 3.1. 3, 9, 16, 22, 31, 36. Section 3.3: 2, 3, 9. Section 3.4: 1. Section 3.5: 2, 4, 12. Solutions by Josh Genauer.
Week 5: Due Friday, May 8 (in class). Section 4.1: 1. Section 4.2: 1, 3, 8, 14. Section 4.3: 29, 30, 32. Section 4.5: 2, 5, 16. Study for Midterm on Wednesday. Solutions by Josh Genauer.
Week 6: Due Friday, May 15 (in class). Section 4.3: 2, 4, 5, 7, 9. Section 4.5: 3, 6, 13, 14. Solutions by Josh Genauer.
Week 7: Due Friday, May 22 (in class). Section 4.5: 29, 30, 36, 40. Section 4.6: 1, 2, 3. Section 5.4: 1, 2, 3. Solutions by Josh Genauer.
Week 8: Due Friday, May 29 (in class). Section 4.4: 1, 3, 5. Section 5.1: 1, 10. Section 5.2: 1, 2, 3, 5, 9. Section 5.5: 11, 12.
April 1: Definition of binary operation, groups, abelian groups, examples (Z, Q, R, C under addition, and Q*, R*, C* under multiplication, Z/nZ, and (Z/nZ)*). Assignment: Read the preliminaries section of the book for Friday's class.
April 3: Order of elements in a group, order of a group, symmetric group, dihedral group.
April 6: More on symmetric group, cycles and factoring permutations, presentation of the dihedral group. Field.
April 8: Matrix groups, homomorphisms and isomorphisms, properties preserved by isomorphisms, (left) group actions, group actions as homomorphisms from the group to permutations of the set.
April 10: D_6 and S_3 are isomorphic. D_8 is a subgroup of S_4. Definition and examples of subgroups. Kernel and Image of a homomorphism are subgroups. Cyclic groups.
April 13: Subgroups generated by sets. More on cyclic groups and their subgroups. The order of elements of cyclic groups.
April 15: Classifying subgroups of cyclic groups; lattice of subgroups; centralizers and normalizers of sets; examples in the dihedral group. Begin Lagrange's theorem.
April 17: (Left) Cosets, partitioning G into a union of cosets, Lagrange's theorem and examples.
April 20: Normal subgroups, quotient group, normal subgroups and kernels of homomorphisms, the first isomorphism theorem.
April 22: Conjugation, group acting on itself by conjugation, simple groups, the product of two subgroups, the isomorphism theorems.
April 24: Simple groups and the Holder program, transpositions and the alternating group.
April 27: Groups acting on sets, orbit, kernel, stabilizer, |orbit|= index of stabilizer group, associated permutation representation, homomorphism from G to symmetric group of A. Example: G acting on itself by left multiplication. More generally, G acting on left cosets of a subgroup H by left multiplication. Cayley's theorem.
April 29: If p is the smallest prime dividing the order of G then a subgroup of index p is normal. Group acting on itself by conjugation. Class equation. Center of a p-group is non-trivial. Statement of Sylow theorems.
May 1: Proof of Sylow theorems assuming the Lemma: If P is a p-Sylow subgroup of G, and Q is a p-subgroup then Q intersect N_G(P) equals Q intersect P.
May 4: (Guest lecture by Brian Conrad)
May 6: In class Midterm.
May 8: Discussion of Midterm questions, recap of Sylow theorems.
May 11: Application of Sylow theorems to groups of order at most 60.
May 13: Wrap up of groups of order up to 60. Using group action on cosets of a subgroup of small index to produce normal subgroups. Proof that A_5 is simple.
May 15: Proof that A_n is simple for n\ge 5. Begin discussion of direct products. Example of groups of order pq with p< q and p not dividing q-1. Proof that if H and K are normal subgroups of G and H intersect K is just the identity then elements of H commute with elements of K.
May 18: Direct product of groups. Beginning of the structure theorem for finite abelian groups.
May 20: Description of the structure theorem for abelian groups. Elementary divisors and invariant factors. Begin talking about semi-direct product.
May 22: Automorphisms of groups; automorphism group of a cyclic group. Definition of the semi direct product.
May 27: Examples of semi direct products. Groups of order pq. Groups of order 30.
May 29: Classification of Groups of order 30 and groups of order 12. Characteristic subgroups of a group. Normal Sylows are characteristic.
June 1: Proof of the fundamental theorem for finite abelian groups.
June 3: