Constructing the 17-gon
The computations are taken farther than what is in the text.
Gauss showed that one can contruct the 17-gon with ruler and compass. To prove that this is possible,
it suces to construct cos , = 2 , which we can write in terms of the
The Multiplicative Group of Integers modulo p
Theorem. Let p be a prime integer. The multiplicative group F of nonzero congruence classes modulo p
p
is a cyclic group.
A generator for this cyclic group is called a primitive element modulo p. The order of
Homework corrected
First version omitted the very important requirement d(0 ) = 1 from the denition of a
harmonic graph.
1. Chapter 10, Exercise 6.4 (characters of the octahedral group).
2. Chapter 10, Exercise 6.9 (identify simple groups from character t
Plane Crystallographic Groups with Point Group D1 .
This note describes discrete subgroups G of isometries of the plane P whose translation lattice L contains
two independent vectors, and whose point group G is the dihedral group D1 , which consists of th
Row Rank = Column Rank
This is in remorse for the mess I made at the end of class on Oct 1.
The column rank of an m n matrix A is the dimension of the subspace of F m spanned by the columns
of A. Similarly, the row rank is the dimension of the subspace of
The Spectral Theorem for Hermitian Matrices
This is the proof that I messed up at the end of class on Nov 15.
For reference: A Hermitian means A = A .
P unitary means P P = I .
Theorem. Let A be a Hermitian matrix. There is a unitary matrix P such that A
Acknowledging Sources in Mathematics Papers
Unless you cite a source, you imply that all wording and results in your paper are yours. If your paper includes
information taken from elsewhere, then you must acknowledge the source of the information. This do
Algebraic Combinatorics
In-Class Exam # 2
April 17, 2009
Open notes. Closed Friends and Enemies. No calculators, computers, I-pods, or
Zunes. Please explain your reasoning or method, even for computational problems.
You may do the problems in any order. T
Algebraic Combinatorics
In-Class Exam # 1
Friday The Thirteenth, March 2009
Open books. Closed Friends and Enemies. No calculators, computers, I-pods, or
Zunes. Please explain your reasoning or method, even for computational problems.
You may do the prob