Math 109 - Applied Group Theory - Winter 2004

| General Info | Syllabus | Announcements & Dates | Homework | Solutions & Handouts |

General Information

Meeting Time Mon., Wed., Fri., 11:00 - 11:50
Location Building 380 (Math), room 380-F
Professor Ben Brubaker (
Office: 382-F (2nd floor, Math building 380)
Office Phone: 3-4507
Office Hours: Monday 2-3:30, Wednesday 2-3:30, or by appointment.
Baosen Wu (
Office: 380-R (Basement, Math building)

Textbook Groups and Symmetry, by M. A. Armstrong.
Midterms -- 20 % each, Final Exam -- 35%, Homework -- 25% (15% problem sets, 10% writing assignment)
We will follow Armstrong's book closely and cover most of the sections over the course of the quarter. We'll begin by introducing the notion of a group and then exploring many examples and subtleties of this definition. Some of the topics to be covered include subgroups, permutation groups, matrix groups, Lagrange's theorem, homomorphisms and isomorphisms, and lattice groups. We will conclude by discussing some of the applications of these concepts to physics, chemistry, and number theory. The evolving syllabus will contain further information.
Prerequisites There are essentially no prerequisites for this course. We will build up to most of the more difficult concepts together in lecture. However, a decent background in linear algebra will be useful in dealing with several of our examples. Some familiarity with proofs and abstract mathematical thinking is a bonus, but not a requirement. Several evening supplementary sessions (completely voluntary) at the beginning of the quarter will provide the essential information on proofs and proof techniques for those who are unfamiliar with them.


Date Material covered Relevant Reading

Wed., 1/7

Introduction, Symmetries of the Tetrahedron

    Armstrong, Ch. 1,
Livio, "The Golden Ratio" (optional)

Fri., 1/9

Definition of a Group and Examples

Armstrong, Chs. 2 & 3

Mon., 1/12

More Examples: Modular Arithmetic, Regular Polygons

Armstrong, Chs. 3 & 4

Wed., 1/14

Dihedral Groups, Orders, Subgroups

Armstrong, Chs. 4 & 5

Fri., 1/16

Generators and Relations, Symmetric Groups

Armstrong, Chs. 5 & 6

Mon., 1/19


Wed., 1/21

More on Symmetric Groups

Armstrong, Ch. 6

Fri., 1/23

Homomorphisms and Isomorphisms

Armstrong, Ch. 7

Mon., 1/26

More Isomorphisms

Armstrong, Ch. 8

Wed., 1/28


Fri., 1/30

Finish Isomorphisms

Armstrong, Chs. 2 & 3

Mon., 2/2

Direct Products

Armstrong, Ch. 10

Wed., 2/4

Cosets and Lagrange's Theorem

Armstrong, Ch. 11

Fri., 2/6

Applications of Lagrange

Armstrong, Ch. 11

Mon., 2/9

Equivalence Relations

Armstrong, Ch. 12

Wed., 2/11

Conjugacy and Normal Subgroups

Armstrong, Ch. 14

Fri., 2/13

How to make a Quotient Group

Armstrong, Ch. 15

Mon., 2/16

NO CLASS (Pres. Day)

Wed., 2/18

Examples of Quotient Groups

Armstrong, Ch. 15

Fri., 2/20


Armstrong, Ch. 16

Mon., 2/23

Examples of Homomorphisms

Armstrong, Ch. 16

Wed., 2/25


Fri., 2/27

Applications: Physics

Course Notes (see Handouts), Armstrong Ch. 9

Mon., 3/1

Relativity and Symmetry, Quantum Mechanics

Course Notes

Wed., 3/3

Applications: Symmetry of Molecules, Group Actions

Course Notes, Armstrong, Ch. 17

Fri., 3/5

Representation Theory

Course Notes

Announcements & Dates

  • Important Dates and Class Holidays
    • Monday, January 19th: NO CLASS (MLK, Jr. Day)
    • Sunday, January 25th: Add deadline
    • Wednesday, January 28th: Exam 1 (In-class)
    • Sunday, February 1st: Drop deadline
    • Monday, February 16th: NO CLASS (Presidents' Day)
    • Wednesday, February 25th: Exam 2 (In-class)
    • Thursday, March 18th: Final Exam, 8:30-11:30 AM, (Location to be determined)
  • A sample midterm is now available in the Solutions and Handouts section of the webpage. Solutions to this midterm will be posted early in the week. A review session will also be held on Tuesday evening. Time to be determined by class vote on Monday, 1/26.
  • The homework due on February 6th asks you to complete a writing assignment on Chapter 8, on Platonic Solids. The idea is to practice a different sort of mathematical communication. Take the discussion of Platonic Solids in chapter 8 as a starting point. Discuss the solids and their groups of symmetry and explain the idea of dual solids. Focus on the solids, and don't worry about describing Cayley's theorem. Moreover, don't simply copy the ideas out of the chapter. Read it, digest it, and come up with something better than Armstrong (you know you can do this). Then take it a bit further. Come up with your own ideas for exploration. Questions you'd like to investigate further. Ideas on the uses of group theory relating to Platonic solids. This is only a first attempt at a finished product. Eventually we will add to these over the course of the quarter, exploring other aspects to be added at the end and waxing philosophical about the process of mathematical discovery to be added at the beginning. There is no page limit (above or below) and you need not type it the first time around. Do try to be mathematically precise and clear at all times. If you have questions, ask or send an email.
  • I have finally finished the test grading. Solutions have now been posted (as of Sunday, 1 pm) and can be found in the solutions section of the web page. I was pleased with the exams; they showed great creativity, which has been the aim of the first three weeks of the course. The grades I listed next to them are a rough idea of the curve I would give the test. However, your grade in the class is based on many different factors and I often adjust the overall grade according to your best performances on all written work. The scores can be found at the following link: Midterm I Scores
  • The second incarnation of the writing assignment is due FRIDAY, MARCH 12 and will be the last written homework for the quarter. You were split into two categories according to the strength of your first effort. If you were asked to rewrite this part, this is a chance for you to correct earlier mistakes and use the group theory that we have learned since the first assignment was due. You can also take this opportunity to explore questions you posed in the first paper or ask and try to answer new questions. What we're looking for here is both an ability to describe mathematical concepts in precise language and evidence of individual creative thinking.
      The SAME is true for those asked to continue on to the new assignment. You've been asked to read pages of a book titled the "Surreal Numbers." The idea is that we have presented group theory as a finished product rather than an evolving science (mostly due to time constraints). The surreal numbers give you a chance to explore a mathematical construction on your own - something closer to research. The book is not long, but I really only expect people to read roughly half of the book and again, show evidence of creative thinking together with an ability to exposit mathematics carefully. Any commentary on the book that possesses these qualities will be graded very favorably. Try to be impressive on these and demonstrate your logical thinking skills. Ask me questions if you are unclear about what is expected.
  • The "Surreal Numbers" by Donald Knuth is available on closed reserve at the Math Library on the 4th floor of the math department. You can check it out on a one-day loan period. There are 4 copies available, so this should be enough for the dozen people asked to write on this topic.
  • Additional information and important class updates can be found here in the future. Examples of this include selected solutions to homework and exams, notes building on in-class discussions, and important changes to the course guidelines. These will ALWAYS be announced in class as well.


Homework assignments are due in class each Friday (except for Friday, January 9th), unless otherwise noted. Reading associated to these homework assignments is also listed. Reading should be done as these chapters are covered in class as a way to review the lectures. Future homework assignments will usually be posted at least one week in advance of the due date.

Assignment Due date Relevant Reading
Chapter 1:
  • 1.2, 1.5, 1.6
    • Chapter 2:

      • 2.2, 2.3, 2.6, 2.7, 2.8
        • Chapter 3:

          • 3.1 (i) & (iii) only, 3.2, 3.3, 3.6, 3.7
Due: Friday, January 16th From Armstrong:
  • Wednesday 1/7: Chapter 1
  • Friday: Chapter 2
  • Monday: Chapter 3
  • #2
    Chapter 4:
    • 4.2, 4.4, 4.6, 4.8, 4.9
      • Chapter 5:

        • 5.1(Z/12Z, D4 only), 5.4, 5.7, 5.10, 5.11
          • Chapter 6:

            • 6.1, 6.3, 6.4, 6.7
    Due: Friday, January 23rd From Armstrong:
  • Wednesday 1/14: Chapters 4 & 5
  • Friday: Chapters 5 & 6
  • #3
    Chapter 6:
    • 6.9
      • Chapter 7:

        • 7.1, 7.4, 7.5, 7.7, 7.8, 7.9
    Due: Friday, January 30th From Armstrong:
  • Wednesday 1/21: Chapter 6
  • Friday: Finish Ch. 6
  • Monday: Chapter 7
  • #4
    Writing Assignment on Ch. 8 (see Announcements)
    Chapter 7:
    • 7.12
      • Chapter 10:

        • 10.2, 10.4, 10.5, 10.7
    Due: Friday, February 6th From Armstrong:
  • Friday 1/30: Finish isomorphisms
  • Monday: Products, Ch. 10
  • #5
    Chapter 11:
    • 11.2, 11.3, 11.4, 11.7, 11.9
      • Chapter 12:

        • 12.2, 12.3, 12.7, 12.8, 12.10
    Due: Friday, February 13th From Armstrong:
  • Wednesday 2/4: Chapter 11
  • Friday: Finish Ch. 11
  • Monday: Chapter 12
  • #6
    Chapter 14:
    • 14.3, 14.5
      • Chapter 15:

        • 15.1, 15.2, 15.6, 15.7, 15.10, 15.12
    Due: Friday, February 20th From Armstrong:
  • Wednesday 2/11: Chapter 14
  • Friday: Begin Ch. 15
  • Wednesday: Finish Ch. 15
  • #7
    Chapter 16:
    • 16.2, 16.3, 16.7, 16.8
      • Chapter 9:

        • 9.2, 9.4
      Due: Friday,
      March 5
    From Armstrong:
  • Monday 2/23: Chapter 16
  • Friday: Begin Ch. 9
  • Monday: Finish Ch. 9
  • #8
    Writing Assignment (see Announcements)
      Due: Friday,
      March 12
    On Closed Reserve in Math Library (380, 4th Floor):
  • The Surreal Numbers, by D.E. Knuth
  • Solutions and Handouts