Conjugacy classes in GL(2,F_q) Notes handed out 5/4/05 by Harold Cooper, listing the conjugacy classes and calculating their sizes. Three page pdf file 5/18/05.
Introduction to Galois theory. Two page pdf file 5/17/05 from Oleg Shamovsky.
Orthogonal geometry. Three page pdf file introducing orthogonal groups. Revision 5/17/05 of notes handed out 5/11 by Oleg Shamovsky.
General parabolic subgroups of Sp(2n). Three page pdf file. Revision 5/17/05 of notes handed out 4/27 by Oleg Shamovsky.
Maximal parabolic subgroups in O(n). Six page pdf file. Revision 5/17/05 of notes handed out 4/21 by Oleg Shamovsky.
Filtered algebras, graded algebras, and Clifford algebras Notes by Yaim Cooper, as handed out in seminar 5/3/05.
Generators for symplectic groups. Description of symplectic transvections, and the proof that they generate the symplectic group; notes by Yaim Cooper on 5/11/05 from her presentation 4/4/05.
Generators for unitary groups. Description of unitary transvections and pseudoreflections, and the proof that they generate the unitary group; notes by Yaim Cooper on 5/11/05 from her presentation 4/8/05.
General parabolic subgroups of GL(n) Three page pdf file from Matt Garcia. Revised 5/16/05.
Classical linear groups are manifolds Five page pdf file from Matt Garcia, describing manifold structures on classical linear groups over the real numbers. Revised 5/11/05 from notes handed out in seminar.
Counting points in O(n)/maximal parabolic Three page pdf file from Katherine Dalis explaining how to count isotropic subspaces for quadratic forms over finite fields. 5/10/05.
Some special elements of orthogonal groups Three page pdf file from Katherine Dalis, describing a special conjugacy class of elements of orthogonal groups. 5/10/05.
Hyperbolic planes and Witt's cancellation theorem for symmetric forms Seven page pdf file from Yelena Yasinnik, featuring a rare photo of a crocheted hyperbolic plane. 5/10/05.
Clifford algebras. Three page pdf file from Eitan Reich 5/9/05.
Counting points in Sp(2n)/maximal parabolic. Five page pdf file from Genevieve Hanlon. Revised 5/5 from handout in class 4/22/05.
Sylow subgroups of Sp(2n,F_q) Three page pdf file from Rachel Lee. Slightly revised and improved from handout in class 4/29/05. Rachel shows that the Sylow subgroup (of order q^{n^2}) has an abelian normal subgroup of order q^{n(n+1)/2}. This suggests a question (for literature search or maybe even do-it-yourself: suppose G is a group of order p^m (with p prime). Is there an abelian normal subgroup N of G of order at least p^{m/2}? (This is true of the Sylow p-subgroups of all the classical groups over F_q, I think.) Revealing the answer could be a nice short presentation.
Simplicity of PSp(2n,F_q).Two page pdf file from Gary Sivek outlining the proof from the text that the projective symplectic group is (almost always) simple.
Hermitian forms. Two page pdf file from Harold Cooper (4/4/05).
PGL(2) over small fields. Three page pdf file from John Rogers about relating PGL(2) over small fields with symmetric groups. Revised 5/4/05.
The symmetric group. Three page pdf file from Harold Cooper.
Maximal parabolic subgroups for the symplectic group. Six page (still incomplete) pdf file from David Vogan. Defines maximal parabolic subgroups for the symplectic group, and sketches a few of their main properties.
The subgroup Omega for the orthogonal group. Eleven page pdf file from David Vogan. The goal (probably not achieved, sadly) was to motivate the definition of the normal subgroup Omega of a special orthogonal group. I prove that if V is an isotropic three-dimensional orthogonal group over a field F, then SO(V) is isomorphic to PGL(2,F); and I define Omega in general.
Order of the finite symplectic group. Seven page pdf file from David Vogan. The main topic is the structure of the stabilizer in a symplectic group of a line; as a consequence, I get the cardinality of the whole symplectic group.
Recursion formula for #G(k,n)(F_q) Four page pdf file from Nicholas Lucero. Posted 3/16/05, as handed out in class.
Sylow theorems for GL_n(F_q). Three page pdf file from Gabe Cunningham. Posted 3/13/05; slight emendation of handout from class.
Classification of quadratic forms over finite fields. Four page pdf file from David Glasser. Posted 3/7/05.
Introduction to bilinear forms. Four page pdf file from Eitan Reich about bilinear forms: definitions of discriminant, radical, and orthogonal; symmetric and alternate forms.
Simple groups. Six page pdf file from Genevieve Hanlon and Rachel Lee about the definition of simple groups, the Jordan-Holder theorem, and Iwasawa's theorem (a method for proving a group is simple). Posted 2/25/05 (slightly revised from the handout in class).
Finite fields. Four page pdf file from John Rogers about finite fields, including some examples of how to construct them explicitly. Revised 5/4/05.
The finite general linear group. Three page pdf file from Gabe Cunningham. Posted 2/24/05.
Grassmann varieties. Five page pdf file from David Vogan about the definition of Grassmann varieties, and how to count points in them over finite fields. Posted 2/21/05.
Projective space. Two page pdf file from David Glasser about the definition of projective space. Revised 2/21/05.
Adjoining roots of a polynomial. Three page pdf file from Gary Sivek describing how to enlarge a field by adding to it a root of a polynomial. Posted 2/9/05. Revised 5/13/05.
Orbits and counting formulas. Three page pdf file describing the basic counting formula for group actions (stated in the seminar on Friday 2/4). Posted 2/7/05.
Tables. One page file with a few multiplication tables for groups. This is more of a typesetting exercise than mathematics. Posted 2/4/05.
Groups. Four page pdf file recalling the definitions of groups and group actions and a few examples. Posted 2/3/05.
Fields. Five page pdf file recalling the axioms for fields and a few examples. Posted 2/1/05.
The Euclidean algorithm. Three page pdf file describing the Euclidean algorithm for computing inverses in Z/nZ. Posted 2/4/00.
Affine subspaces. Two page postscript file defining affine subspaces of vector spaces, and sketching some basic properties of them. Posted 2/15/00.
My TeX style isn't particularly admirable, but it's always nice to have something to copy when you want to do something tricky. I'll be encouraging you to prepare TeX files of notes for your seminar presentations, so you may want to look at the tex files for what's here. At least some of those files (usually AMS-TeX) will be here, but without links; you can guess the addresses from those for the corresponding posted file. (For example, the TeX file for http://www-math.mit.edu/~dav/fields.pdf is at http://www-math.mit.edu/~dav/fields.tex)