Math 67

General information

Give me some anonymous feedback! (Be sure to state somewhere in the comments that the feedback is for Math 67 or else I might end up improving a different course!)

Guidelines for "textbook style" proof writing

• a sample document (the LaTeX file that created the guidelines for proof writing, above)
• some commonly-used mathematical symbols
• the source for everything TeX!
• If you are new to LaTeX, I highly recommend signing up for a (free) account with Overleaf (formerly, WriteLaTeX). They have lots of templates that you can modify directly to create your own document or that you can peruse to get a feel for how to use LaTeX.
• This is a nice source for examples of how to create graphics with tikz. (You won't need diagrams for a while.)
• Some mapping diagrams and the tikz code that creates them can be found here.

Announcements

Office hours for the next two weeks can be found here.

Solutions to the exercises in the textbook will be on reserve in Cornell as of Thursday, 12/14

Fill out an end-of-the-course assessment for extra credit! (Check your email for link to survey.)

The final exam is Tuesday, 12/19, from 2 - 5 pm, in Sci 181.

Homework

#0, due Wednesday, 9/6: Reading: Section 0; Problems: Section 0 -- 1, 2, 11, 12, 16d, 26, 29, 30, 31, 32 (for 29 - 32, ignore the part about describing the partition); this problem; automathography; problem session survey. Please follow the writing guidelines when giving your solutions to 29, 30, 31, 32, and the extra problem. In #12, if the given relation is not a function or is not one-to-one or onto, state why. For the rest of the problems, you may just write down the answer.

#1, due Wednesday, 9/13: Reading: Section 2; Problems: Section 2 -- 2, 4, 7, 8, 18, 19, 20, 22, 23, 34, 36, 37; these extra problems; Section 1 -- 6, 10, 13, 41. Please follow the writing guidelines on problems 2.23, 2.36, 2.37, and the three extra problems. The rest of the problems are exempt from the writing guidelines, but your answers to them should nonetheless be clear and complete, with easy-to-follow logic. (You may use abbreviations, symbols, and incomplete sentences in your solutions to the exempt problems.) Please fill out this problem session survey by Monday at noon.

#2, due Wednesday, 9/20: Reading: Section 3 and Section 4 up until "finite groups and group tables" on p. 43; Warning! This week's assignment is LO-O-O-NG! Problems: Section 3 -- 3, 4, 8, 15, 30, 32; Section 4 -- 6, 8, 10a, 12, 14, 18, 29, 30, 32, 34, 38; these problems. The problems in bold, plus the 2nd extra problem, are the ones that need to be written up in "textbook style"--that is, using the writing guidelines. Please fill out this problem session survey by Monday 9/18 at noon.

#3, due Wednesday, 9/27: Reading: the rest of Section 4 and all of Section 5; Problems: Section 4 -- 41; Section 5 -- 4, 8, 15, 16, 22, 34, 39efi, 41, 43, 45, 48, 52, 53, 54; this problem. The problems in bold, plus the extra problem, are the ones that need to be written up in "textbook style"--that is, using the writing guidelines. Please fill out this problem session survey by Monday 9/25 at noon. With the exception of problems 22, 34, and 39, none of the problems require any algebraic knowledge outside of the definition of a subgroup (though you may find it more efficient to use Theorem 5.14 to prove something is a subgroup rather than going back to the definition itself for the proof).

#4, due Wednesday, 10/4: Reading: Section 6; Problems: Section 5 -- 57; Section 6 -- 2, 6, 18, 23, 28, 32abdefgi, 36, 44, 49, 52, 55 . The problems in bold, plus the extra problems, are the ones that need to be written up using the writing guidelines. Please fill out this problem session survey by Monday 10/2 at noon.

#5, due Wednesday, 10/11: Reading: Section 8; Problems: Section 8 -- 1, 2, 3, 5, 6, 8, 9, 12, 16, 35, 40, 45, 47 (as of Wednesday, 10/4 , we have covered everything necessary to solve all the problems except 12 and 35); these problems. Only The only problems that need to be written up in "textbook style" are extra problems 1, 2a, 3, and 5. Please fill out this problem session survey by Monday, 10/9, at noon. A reminder about problem session expectations: on all but the most exceptional weeks, you are expected to come to problem session having solved, and willing to present, at least half of the problems. Your work in problem session counts as 15% of your final course grade!

#6, will not be collected!: Reading: Section 9; Problems: Section 9 -- 1, 3, 5, 9, 11, 13, 15, 17, 27ab, 29, 33. Here are pages from the solution manual that cover the proof questions in this non-assignment. (The other answers can be found in the back of your book.) Problem session on Monday, 10/23, is optional, and will be run as office hours/a review session for the midterm Tuesday night. Bring your questions!

#7, due Wednesday, 11/1: Reading: Section 10; Problems: Section 10 -- 1, 4, 14, 16, 19abcghij, 20, 24, 28, 29 (despite what the book says, this is not exactly the same as the converse of #28; you will first need to explicitly show why the hypothesis in #29 implies that every left coset gH is the same as the right coset Hg), 30, 31, 34, 39, 40. The only problems that need to be written up in "textbook style" are the problems in bold. Please fill out this problem session survey by Monday, 10/29, at noon.

#8, due Wednesday, 11/8: Reading: Sections 11 & 13; Problems: Section 11 -- 6, 14, 15, 16, 18, 24, 32, 36, 37, 46, 51, 52; Section 13 -- 1, 2, 6, 8, 9, 10, 18, 21; this problem. Note: as always, you only need to follow the writing guidelines for problems that are in bold. Please fill out this problem session survey by Monday, 11/6, at noon.

#9, due Wednesday, 11/15: Reading: Section 14; Problems: Section 13 -- 32, 47, 49, 50; Section 14 -- 6, 10, 12 (and determine which group of order 4 this factor group is isomorphic to), 23abcdghi, 30, 31, 34 (Hint: what are the possible conjugate subgroups of H?), 35, 39 (don't forget to show the map $\phi_*$ is well-defined!), 40b (even though you showed this was a subgroup many moons ago, redo it here, citing facts from linear algebra); Note: as always, you only need to follow the writing guidelines for problems that are in bold. Please fill out this problem session survey by Monday, 11/13, at noon.

#10, due Wednesday, 11/22: Reading: Sections 15 & 18; Optional: Here is a proof that the alternating groups are simple for n>=5. Problems: Section 15 -- 2, 3, 4, 19, 30, 31, 35, 40, 41; Section 18 -- 8, 11, 22, 33, 37, 41, 43, 48, 50; Note: as always, you only need to follow the writing guidelines for problems that are in bold. Please fill out this problem session survey by Monday, 11/20, at noon.

#11, due Wednesday, 11/29: Reading: Sections 19 & (through the top half of p. 187 of) 20. Problems: Section 19 -- 7, 10, 17, 18, 25, 29; Section 20 -- 2, 4, 10, 23abcdefgh, 27; Section 21 -- 4abcdehj; Note: as always, you only need to follow the writing guidelines for problems that are in bold. Please fill out this problem session survey by Monday, 11/27, at noon.

#12, due Wednesday, 12/6: Reading: Sections 22 & 23; Problems: Section 22 - 2, 6, 10, 12, 14, 21, 22, 23, 25, 27; Section 23 -- 6, 10, 12, 25, 28, 34, 35, 36 (Long assignment, but mostly computational, not proofs). Please fill out this problem session survey by Monday, 12/4, at noon.

#13 is for (optional!) extra credit, due Wednesday, 12/13 at noon outside my office: Reading: Sections 23 & 26; Problems: Section 23 - 14, 16, 17, 18, 20; Section 26 -- 3, 4, 9, 10, 12, 13, 14, 15, 18, 20, 22ab, 26, 29

Practice problems from Section 27: 1, 2, 3, 15, 16, 17, 24, 31 (not to be turned in with optional assignment #13)