Ken Brown - Notes

This page contains materials that I've prepared for various courses I've taught. I'm collecting the materials here in the hope that they might be useful for other teachers. Feel free to use them (with proper attribution), and please send me a note indicating that you have done this. Also let me know if you find any mistakes or broken links.

Mathematics 7350, Fall 2013, Computational group theory

Handouts

• The Todd–Coxeter procedure
• Todd–Coxeter example (beamer presentation)
• Computation in permutation groups: Introduction
• Reverse Todd–Coxeter: Example
• The Knuth–Bendix procedure: Example

GAP demos

• Interactive Todd—Coxeter on <a,b,c ; a^c = a², b^a = b², c^b = c²>
• Random permutation group
• Sylow subgroups
• The demo related to the reverse Todd–Coxeter example did several coset enumerations using the ITC package. Here are the corresponding GAP files:
  • Initial definitions
  • The relation c⁴ = 1
  • Add the relation ab^-1 = c²
  • Add the relation a²b = c
  • Add the relation (ac)² = 1
• The demo related to the Knuth–Bendix example did two computations using the KBMAG package. Here are the corresponding GAP files:
  • Initial attempt that doesn't terminate
  • Succesful run with big rules discarded
• As I explain at the end of the handout on the Knuth–Bendix example, the choice of generators in the group G of that example is based on the nilpotent quotient algorithm. The NQ package for GAP produces the maximal nilpotent quotient H of G, as shown here:
  • Nilpotent quotient example
  In order to actually see the definitions of the generators of H, however, you have to run the standalone nq program on the following file:
  • A presentation of G
• The demo on automatic groups did two computations using the KBMAG package. Here are the corresponding GAP files:
  • A hyperbolic group
  • The Heineken group
• The Neubüser–Sidki group, used to illustrate the nilpotent quotient algorithm.
• The low-index subgroups algorithm:
  • The 2-3-7 triangle group
  • The Heineken group

Mathematics 4530, Fall 2013, Introduction to topology

• Ordinal numbers and the well-ordering theorem
• Tychonoff's theorem

Mathematics 6310, Fall 2011, Algebra

• The butterfly lemma
• Introduction to category theory
• Solution to Exercise 50 on p. 148
• The Todd–Coxeter procedure
• Computation in permutation groups
• Zorn's lemma
• Ordinal numbers
• The primitive element theorem

Mathematics 3360, Spring 2010, Applicable algebra

• Computing powers mod m
• The primitive root theorem
• Separable polynomials

Mathematics 4340, Spring 2009, Honors introduction to algebra

• The number of groups of order n (extracted from Besche, Eick, and O'Brien, A millennium project: constructing small groups, International Journal of Algebra and Computation 12 (2002), no. 5, 623–644). The complete article is available online via the Cornell library catalogue. For theoretical results on the number of groups of order n, see Blackburn, Neumann, and Venkataraman, Enumeration of finite groups, Cambridge University Press, 2007.
• Simplicity of the alternating groups
• A nonabelian finite simple group has a proper nonabelian subgroup
• What is a free group?
• What is a group presentation?
• The Todd–Coxeter procedure
• When are all groups of order n cyclic?
• A presentation for the symmetric group
• On the relation bab^-1 = a²
• Root adjunction
• Zorn's lemma
• The primitive element theorem

Mathematics 634, Spring 2008, Commutative algebra

• The Galois correspondence between a ring and its spectrum
• Remarks on artinian rings
• Remarks on associated primes