MATH 3560 is a geometric introduction to the algebraic theory of groups, through the study of symmetries of planar patterns and 3–dimensional regular polyhedra. Besides studying these algebraic and geometric objects themselves, the course also provides an introduction to abstract mathematical thinking and mathematical proofs, serving as a bridge to the more advanced 4000–level courses. Abstract concepts covered include: axioms for groups; subgroups and quotient groups; isomorphisms and homomorphisms; conjugacy; group actions, orbits, and stabilizers. These are all illustrated concretely through the visual medium of geometry.

MATH 3560 is a 4–credit course. The course catalogue states that the prerequisite is at least one of

MATH 2210 — Linear Algebra

MATH 2230 — Theoretical Linear Algebra and Calculus

MATH 2310 — Linear Algebra with Applications

MATH 2940 — Linear Algebra for Engineers.

If you have not satisfied the prerequisite, seek permission of the instructor to take the course. Prior knowledge of group theory is not a prerequisite.

Grades and academic integrity

Grades will be computed according to:

Prelim

15%

Essay

15%

Presentation

5%

Final exam

35%

Homework

30%.

Cornell's code of academic integrity applies to this and all other courses. In particular, academic misconduct of any kind may result in a grade penalty or the assignment of a failing grade.

Examinations

Prelim

In class, 8th October 2013, covering Chapters 1–8

The exam (scored out of 40), Solutions, Mean: 29.2, Median: 28.3, Standard deviation: 4.6

Final

7:00pm–9:30pm, Wednesday 18th December 2013, RCK110, comprehensive

Exams will be taken without the aid of any textbooks or notes.

Calculators will NOT be allowed for the exams. Cell phones, MP3 players and other electronic devices (aside from watches) will NOT be allowed in the exam room, not even as time–keeping devices.

Accommodations may be made in the event of illness, a clash with another examinations, a conflict with religious observances, or other serious circumstances — please contact your instructor as early as possible. Students who miss exams and then present excuses that could have been given in advance will not be looked upon favorably. Do not make travel plans that conflict with the exams.

If you have been approved for extended–time or special–condition exams by Cornell Student Disability Services, notify your instructor in the first two weeks of the semester.

Due 9/3:

Make a tetrahedron.

1.3, 1.4, 1.9

Due 9/10:

Draw the Cayley diagram (a.k.a. Cayley graph) for the rotational symmetries of the tetrahedron with respect to the rotations r and s of Ex. 1.3 and 1.4.

2.2, 2.3, 2.8, 3.5, 3.9, 3.10

Due 9/17:

4.2, 4.5, 4.8, 5.3, 5.8, 5.9

Due 9/24:

6.1, 6.2, 6.4, 6.9, 6.11, 6.12

Draw the Cayley diagram for S4 with respect to (1 2 3 4) and (1 2).

Due 10/1:

Make a cube or octahedron, and a dodecahedron or icosahedron. (A prize for the best!)

What do you get if you join up the midpoints of adjacent faces of a tetrahedron? What if, instead, you join up the midpoints of adjacent edges?

7.1, 7.3, 7.6, 8.1, 8.4

10/8:

No homework (prelim)

Due 10/17 (note day):

9.1, 9.5, 9.7, 9.12, 10.6, 10.7, 10.11

Due 10/22:

11.1, 11.2, 11.3, 11.4, 11.8

Due 10/29:

12.2, 12.3, 12.7, 12.8, 12.12 (read Example vi in Chapter 12), 13.2, 13.4

Due 11/5:

14.1, 14.4, 14.5, 14.6, 14.11, 15.3, 15.6, 15.12

11/12:

Please work on your essays! Hand in a one-page essay plan (not for credit).

Due 11/19:

16.1, 16.7, 16.8, 16.10, 17.2, 17.3, 17.9 (C was defined in Chapter 3), 17.10

11/26:

No homework (essays due)

Due 11/3:

18.2, 18.3, 18.5, 19.2, 19.3, 19.4, 19.7

Solutions. Follow this link to TA's page for solutions and statistics.

Practical information. MATH 3560 is a four–credit course, so you should plan on spending at least twelve hours a week working on it. Homework is the most important part of MATH 3560 since mathematics is really learnt by doing it.

Homework assignments will be due in class on Tuesdays. Papers should be stapled and your name clearly indicated. Late homework will not be accepted. If you have a medical excuse or other serious circumstances, allowences will be made; contact your instructor at the earliest opportunity. The lowest score across the semester will be dropped (as everyone has a bad week sometime).

I expect homework to be graded and returned within a week. Due to constraints on resources, not all problems may be graded.

Collaborating. You may collaborate with other students on homework. I believe, however, that for maximum benefit, you should try hard to do all the problems yourself before consulting others. What you turn in should represent your own solutions expressed in your own words, even if you arrived at these solutions with others. Remember, you are doing the homework to learn the material; do not try to defeat its purpose. Copying someone else's homework and presenting it as your own will be treated as a violation of Cornell's Academic Integrity Code, as will copying solutions that you might find on the internet or elsewhere.

In keeping with the good practice of acknowledging all contributors to a piece of work, if you do collaborate, please give the names of your collaborators so on your homework. (Your grade will not be affected.)

Students agree that by taking this course all required papers may be subject to submission for textual similarity review to Turnitin.com for the detection of plagiarism. All submitted papers will be included as source documents in the Turnitin.com reference database solely for the purpose of detecting plagiarism of such papers. Use of the Turnitin.com service is subject to the Usage Policy posted on the Turnitin.com site.

Writing well. Mathematics is not mere computation; arguments and abstract concepts must be communicated. Use complete sentences with proper spelling, grammar, and punctuation. Write linearly down the page rather than scattering words, symbols and equations around. Explain your reasoning carefully. Indicate the significances of and relationships between any calculations which contribute to your answers. Define and employ clear and concise notation. State clearly any results (from lectures or from the textbook) to which you appeal in your solutions. Imagine your fellow-students as your readers — ask yourself whether they would they be able to follow your arguments.

Here is some helpful guidance on communicating mathematics. It was developed by Tara Holm from an earlier version by Peter Kahn, a previous instructor of this course.

If you would like to try using LaTeX for typing up your homework (to quote the above document "it beats the pants off Word"), here's a guide to getting started. Whilst LaTeX produces beautiful results and is a good skill to have, it'll take a lot of work at first. Well hand-written homework is fine instead.

Support. The Mathematics Support Center in Malott 256 provides free tutoring. Details of my office hours and those of the T.A. are on the course homepage.