Lecture notes on the Web
This list gives you access to lecture notes in design theory,
finite geometry and related areas of discrete mathematics on
the Web. I have given a brief annotation and table of contents
for each set of notes.
Note on formats: HTML files should be handled
by your browser. Others require special software to display or print them.
On my system, these are xdvi (for DVI), ghostview (for
PostScript), and acroread (Acrobat reader for PDF). All these are
freely available and can be wired into your browser.
Please email me with
further information for this section. I will include readers' reviews
if anyone is prepared to submit them!
Contents of this page:
 Socrates Intensive Courses on Finite Geometry and
Applications
 Lecture notes
on permutation groups
 Individual sets of notes:
 Ian Anderson and Iiro Honkala,
A short course in combinatorial designs
 R. A. Bailey,
Association schemes and partially balanced designs
 A. Betten, H. Fripertinger and A. Kerber,
Algebraic combinatorics via finite group actions
 Chris K. Caldwell,
Graph theory tutorials
 Peter J. Cameron,
Classical groups
 Peter J. Cameron,
Projective and polar spaces
 Peter J. Cameron,
Polynomial aspects of codes, matroids and permutation groups
 Bill Cherowitzo,
Combinatorial structures
 Queen Mary Combinatorics Study Group Papers
 Stephen Donkin,
Linear Algebra
 Andrew Granville,
Arithmetic properties of Binomial Coefficients
 W. D. Joyner,
The Mathematics of Rubik's Cube

M. Klin et al.,
Algebraic combinatorics in mathematical chemistry

László Lovász's Lecture Notes
 T. W. Müller,
Five lectures on generalized permutation representations
 Steven R. Pagano,
Matroids and signed graphs
 John Preskill,
Quantum information theory and quantum computing
 D. R. Stinson,
Combinatorial designs with selected applications
 Herb Wilf,
East side, west side
 Other sources of notes
There are several sets of lecture notes from the Socrates Intensive
Courses in Finite Geometries and Applications.
Ian Anderson and Iiro Honkala,
A short course on combinatorial designs, available from
http://www.utu.fi/~honkala/designs.ps
The basics in less than forty pages.
Format: PostScript
Contents:
 Systems of distinct representatives
 2designs
 tdesigns and Steiner systems
 Codes and designs
R. A. Bailey,
Association schemes and partially balanced designs, available
from
http://www.maths.qmul.ac.uk/~rab/MAS417/
Notes of a course currently in progress. Written
from a statistician's point of view, these notes are complementary to the
treatments by Bannai and Ito or by Brouwer, Cohen and Neumaier: they have
much to say about methods of calculation and about highly imprimitive
association schemes, for example. Also includes an annotated
reading list, and course problem sheets. The notes are adapted from a
forthcoming book.
Format: PDF
Contents:
 Definitions of association scheme
 Adjacency matrices
 Some special association schemes
 The BoseMesner algebra
 Character tables
 Techniques
 Strongly regular graphs
 Block designs
 Partially balanced block designs
 A little statistics
 Efficiency
 Cyclic designs
 Families of partitions
 Orthogonal block structures
A. Betten, H. Fripertinger and A. Kerber,
Algebraic combinatorics via finite group actions, available from
http://bedvgm.kfunigraz.ac.at:8001/frib/html2/book/hyl00.html
A very complete survey of enumeration under group
action; in interactive HTML, so you can try out the concepts for
yourself. Many exercises.
But you have to reconfigure your Xwindows to get the symbols to
print correctly.
Format: HTML
Contents:
 Actions (Actions of groups; Bilateral classes, symmetry classes of
mappings; Finite symmetric groups; Complete monomial groups;
Enumeration of symmetry classes; The involution principle;
Special symmetry classes)
 Weights (Enumeration by weight; Cycle indicator polynomials;
Sums of cycle indicators, recursive methods; A generalization;
The Decomposition Theorem; Species)
 Marks
 Constructions (Orbit evaluation; Transversals of symmetry classes;
Orbits of centralizers; Recursion and orderly generation;
Generating orbit representatives; Symmetry adapted bases)
 Index
Arrigo Bonisoli, On collineation groups of
finite planes, (Socrates course notes), available from
http://cage.rug.ac.be/~fdc/intensivecourse2/bonisoli2.pdf
The heart of Dembowski's book Finite Geometries,
published in 1968, concerned collineations of finite projective planes.
This material has never been satisfactorily brought up to date. These
notes do so in part, including a lot of recent material on collineation
groups fixing an oval.
The addendum, available from
http://cage.rug.ac.be/~fdc/intensivecourse2/bonisoli_koen.pdf,
contain exercises and further explanations.
Format: PDF
Contents:
 Introduction
 The basics
 Some classics
 Machinery
 Primitive ovals in projective planes of odd order
 An excursion into graphs
 The uniqueness of the dual Lüneburg plane of order 64
 Recent results, open problems
Contents of addendum by K. Thas:
 Notation, definitions, and some exercises
 Ovals, ovoids, and a theorem concerning the fixed element structure of
a collineation of a finite projective plane
 Translation nets, translation planes, Moufang planes, and
( p,L)transitivity
 Some easy remarks on the proofs of some propositions
 On Proposition 4.3
 On Section 5
 Moufang sets and collineations of projective planes
 Collineations of finite projective planes and the Petersen graph
 References
Matthew Brown, (Hyper)ovals and ovoids in projective
spaces, (Socrates course notes), available from
http://cage.rug.ac.be/~fdc/intensivecourse2/brown_2.pdf
An oval or ovoid in a finite projective space
of odd characteristic is "classical" (a conic or elliptic quadric), but there
are other examples in characteristic 2, and the classification problems are
still open. This survey gives proofs of the classical results, and uptodate
information in characteristic 2 including the recent "Adelaide ovals".
Format: PDF
Contents:
 Introduction
 (Hyper)ovals in PG(2,q)
 Ovoids in PG(3,q)
 References
Francis Buekenhout, History and prehistory of
polar spaces and of generalized quadrangles, (Socrates course notes),
available from
http://cage.rug.ac.be/~fdc/intensivecourse2/buekenhout3.pdf
The story of one of the most important topics in finite
geometry, told by one who contributed more than anyone else except Jacques
Tits to the story.
Format: PDF
Contents:
 Introduction
 Projective spaces
 Duality and polarity
 The classical groups, Jordan 1870 and Dickson 1901
 Tits 1956
 Veldkamp 1959
 Tits 1959, FeitHigman 1964
 Tits 19611974
 BuekenhoutShult 1974
 To be done
 References
Chris K. Caldwell, Graph theory tutorials, available
from
http://www.utm.edu/departments/math/graph/
Created with WebTutor; you have to register, and
you have to do the exercises before you are allowed to proceed. Aimed at
beginning graph theorists.
Format: HTML (CGI)
Contents:
 Introduction to graph theory
 Euler circuits and paths
 Coloring problems
(More tutorials are promised)
Peter J. Cameron,
Finite geometry and coding theory,
(Socrates course notes), available from
http://dwispc8.vub.ac.be/Potenza/lectnotes.html
Mainly about quadratic forms over GF(2) and their role in
finite geometry, classical and quantum errorcorrection,
extraspecial groups, etc. Includes exercises.
Format: PostScript
Contents:
 Codes
 Symplectic and quadratic forms
 ReedMuller codes
 Selfdual codes
 Bent functions
 Kerdock codes
 Some resolved designs
 Extraspecial 2groups
 Quantum computing
 Quantum codes
 Z_{4} codes
 Bibliography
Peter J. Cameron,
Classical groups,
available from
http://www.maths.qmul.ac.uk/~pjc/class_gps/.
Notes of a lecture course, roughly following Taylor's book:
generation and simplicity of classical groups, and some of
their geometry. Includes exercises.
Format: DVI, PostScript, PDF
Contents:
 Fields and vector spaces
 Linear and projective groups
 Polarities and forms
 Symplectic groups
 Unitary groups
 Orthogonal groups
 The Klein correspondence and triality
 Further topics
 A short bibliography on classical groups
Peter J. Cameron,
Projective and polar spaces,
available from
http://www.maths.qmul.ac.uk/~pjc/pps/.
Second edition (with revisions and corrections) of QMW Maths Notes 13,
first published in 1991. Describes the geometry of projective and polar
spaces, with extras such as Clifford algebras, Mathieu groups, and
diagram geometry. Includes exercises.
Format: PDF
Contents:
 Projective spaces
 Projective planes
 Coordinatisation of projective spaces
 Various topics
 Buekenhout geometries
 Polar spaces
 Axioms for polar spaces
 The Klein quadric and triality
 The geometry of the Mathieu groups
 Exterior powers and Clifford algebras
 References and index
Peter J. Cameron,
Polynomial aspects of codes, matroids and permutation groups,
available from
http://www.maths.qmul.ac.uk/~pjc/csgnotes/cmpgpoly.pdf.
These notes include background on codes, matroids and permutation groups,
and polynomials associated with them (weight enumerator, Tutte polynomial
and cycle index), and describe the links between these objects. Their
second purpose is to describe codes over Z_{4} and the
associated matroids and permutation groups.
Format: PDF
Contents:
 Codes
 Codes over Z_{4}
 Matroids
 Matroids and codes
 Permutation groups
 Cycle index
 Codes and permutation groups
 IBIS groups
Bill Cherowitzo, Combinatorial structures,
available from
http://wwwmath.cudenver.edu/~wcherowi/courses/m6406/m6406f.html
A combinatorics course, slanted to topics of interest to
design theorists and finite geometers. In welldesigned HTML, with homework
problems.
Format: HTML
Contents:
 Latin squares
 Introduction to finite fields
 Hadamard matrices
 Block designs
 Finite geometries
Queen Mary Combinatorics Study Group Papers,
available from
http://www.maths.qmul.ac.uk/~pjc/csg.html#csgpapers
An occasional series of expository papers about topics
discussed in the Study Group.
Format: DVI, PostScript, PDF
Contents:
 Fibonacci notes, by Peter Cameron and Dima FonDearFlaass
 Notes on quantum error correction, by Harriet Pollatsek and Keldon Drudge
 Problems from the First AngloHungarian Meeting on Groups and Geometries
 Five lectures on generalized permutation representations, by Thomas
Müller
 Borcherds' proof of the moonshine conjecture, after V. Nikulin
 Partially ordered sets, by Thomas Britz and Peter Cameron
 Causal set glossary and bibliography, by Rafael D. Sorkin
 A Markov chain for Steiner triple systems, working notes by Peter Cameron
 Primitive lambdaroots, by Peter Cameron and Donald Preece
 Decoding the Mathieu group M_{12}, by Robert F. Bailey
Stephen Donkin, Linear algebra,
available from
http://www.maths.qmul.ac.uk/~donkin/linearalgebra/
An elementary course on linear algebra. Includes both
the concrete approach via matrices and the abstract approach via vector
spaces.
Format: PDF
Contents:
 Matrices
 Inversion, elementary row operations
 Determinants
 Vector spaces
 Linear maps
 The rank of a matrix
J. Eisfeld and L. Storme,
(Partial) tspreads and minimal tcovers in finite projective
spaces, available from
http://cage.rug.ac.be/~fdc/intensivecourse2/storme2.ps
A survey of this topic, including extendability of
partial spreads to spreads and largest/smallest cardinality of
partial spreads/covers of projective spaces.
Format: PostScript
Contents:
 tspreads of projective spaces
 tcovers and partial tspreads in PG(N,q),
where t+1 does not divide N+1
 Extendability of spreads in PG(3,q)
 Partial spreads in PG(N,q), where t+1 divides N+1
 Minimal tcovers in PG(N,q), where t+1
divides N+1
 General covering and blocking problems in projective spaces
 References
Andrew Granville,
Arithmetic properties of Binomial Coefficients, available from
http://www.math.uga.edu/~andrew/Binomial/index.html
From Lucas' Theorem to recent results on congruences
of binomial coefficients modulo prime powers. This is a dynamic survey
which is expected to develop.
Format: HTML
Contents:
 Introduction
 Elementary Number Theory
 Generalization of Lucas' Theorem
 Congruences for sums of binomial coefficients
 Computing binomial coefficients modulo prime powers
 Recognizing the primes
 Pascal's triangle via cellular automata
 Studying binomial coefficients through their generating function
 Bernoulli numbers and polynomials
 Theorems of Morley and Emma Lehmer and their generalizations
 Some useful padic numbers
 Congruences modulo higher powers of primes
 Concluding remarks
 References
Willem H. Haemers,
Matrix techniques for strongly regular graphs and related geometries,
(Socrates course notes), available from
http://cage.rug.ac.be/~fdc/intensivecourse2/haemers2.pdf
After surveying strongly regular graphs and their
generalisations (distanceregular graphs and association schemes), these
notes discuss matrix techniques (partitions and interlacing), with an
application to the uniqueness of a particular strongly regular graph.
The addendum, available from
http://cage.rug.ac.be/~fdc/intensivecourse2/haemers_extensie.pdf
gives supplementary results, mostly more geometric in
character.
Format: PDF
Contents:
 Strongly regular graphs
 Association schemes
 Matrix tools
 The (81, 20, 1, 6) strongly regular graph
 References
Contents of addendum by S. Cauchie and E. Kuijken:
 Eigenvalues of a regular graph
 The friendship property; polarities in projective planes
 The line graph of a graph
 (Partial) linear spaces and their point and line graphs
 Pseudogeometric graphs
 Spreads
 References
J. W. P. Hirschfeld,
Semilinear groups over finite fields,
(Socrates course notes), available from
http://dwispc8.vub.ac.be/Potenza/lectnotes.html
Discusses the types of polarities of projective spaces and
the semilinear groups they define.
Format: PostScript
Contents:
 Polarities
 Groups on the line
 Orders and isomorphisms among the semilinear groups
 Bibliography
W. D. Joyner,
The mathematics of Rubik's cube,
available from
http://www.permutationpuzzles.org/rubik/webnotes/rubik.pdf
An introduction to the discrete mathematics and group theory
underlying Rubik's cube and other permutation puzzles. A fine
example of nontrivial mathematics arising from "diversions".
Many worked examples and exercises.
Format: DVI
Contents:
 Logic and sets
 Functions, matrices, relations and counting
 Permutations
 Permutation puzzles
 Groups, I
 Graphs and "God's Algorithm"
 Symmetry groups of the Platonic solids
 Groups, II
 The Rubik's cube and the word problem
 The 2 × 2 and 3 × 3 cube groups
 Other Rubiklike puzzle groups
 Interesting subgroups of the cube group
 Crossing the Rubicon
 Appendix: some solution strategies
M. Klin, Ch. Rücker, G. Rücker, G. Tinhofer,
Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms,
I, Permutation Groups and Coherent (Cellular) Algebras,
available from
http://wwwlit.ma.tum.de/veroeff/html/950.05003.html
This valuable exposition and survey brings together
ideas about graph isomorphism, cellular algebras, permutation groups, and
mathematical chemistry.
Format: PostScript
Contents:
 Introduction
 The subject of algebraic combinatorics
 Problems related to the perception of the symmetry of chemical graphs
 Fundamentals of permutation group theory
 Centralizer algebras of permutation groups
 Cellular algebras
 Galois correspondence between permutation groups and cellular algebras
 Srings over cyclic groups
 Automorphism groups of certain chemical graphs
 Concluding remarks
O. H. King,
Classical groups,
(Socrates course notes), available from
http://dwispc8.vub.ac.be/Potenza/lectnotes.html
A general account of classical groups including Aschbacher's
Theorem.
Format: PostScript
Contents:
 Forms and groups
 Isomorphisms between classical groups
 Aschbacher's Theorem
 Bibliography
T. W. Müller,
Five lectures on generalized permutation representations,
available from
http://www.maths.qmul.ac.uk/~pjc/csgnotes/LecBras.ps
Lectures given by the author in an algebra summer school
in Brazil, and repeated in the Queen Mary Combinatorics Study Group. They
describe techniques for counting representations of an arbitrary finitely
generated group in a wreath product H wr S_{n}
(or a variant on this), with applications to such topics as Quillen
complexes and subgroup growth.
Format: PostScript
Contents:
 Some combinatorial aspects of permutation representations
 Generalizing permutation representations
 Some examples and a formula for the exterior function
 Explicit formulae for abelian groups and computations in Quillen complexes
 Asymptotics of
Hom(G, H wr S_{n})
and subgroup growth
László Lovász,
Lecture notes, available from
http://www.cs.yale.edu/homes/lovasz/notes.html
Various sets of notes by Lovász on topics
in discrete mathematics. Packed with insight, clear explanation, and many
exercises. Recommended.
Format: PostScript
Contents:
 Semidefinite optimization
 Topological methods in combinatorics
 Complexity of algorithms
 (with K. Vesztergombi) Discrete Mathematics
F. Mazzocca,
Nuclei in projective planes, available from
http://cage.rug.ac.be/~fdc/intensivecourse2/nuclei_2.ps
A nucleus of a set S of q+1 points in a
projective plane of order q is a point all lines through which meet
S in just one point. These notes tell what is known about nuclei,
and generalisations to affine planes and to "multiple nuclei".
Format: PostScript
Contents:
 Definition and examples
 Some results
 The SegreKorchmáros lemma and its applications
 Sets with the maximal number of nuclei
 Quasiodd sets
 Generalizations
 Further generalizations
 References
Steven R. Pagano,
Matroids and signed graphs, available from
http://www.ms.uky.edu/~pagano/Matridx.htm
A student'seye view of these topics  clearly introduced.
Format: HTML
Contents:
 Introduction and some notes
 What is a matroid?
 Some common examples of matroids
 Circuits, bases, rank, closure
 Duality
 Minors
 Representability
 Connectivity
 What the heck is a signed graph?
 What are you trying to find out?
 Have you found anything interesting yet?
 Any good references for all this stuff?
John Preskill,
Quantum information theory and quantum computation,
available from
http://www.theory.caltech.edu/people/preskill/ph229/
A careful exposition of this important topic, for the Caltech course
PH229. Includes new material on quantum error correction of interest
to design theorists. Includes exercises. Recommended.
Format: PostScript
Contents:
 Introduction and overview
 Foundations of quantum theory I: States and ensembles
 Foundations of quantum theory II: Measurements and ensembles
 Quantum entanglement
 Quantum information theory
 Quantum computation
 Quantum error correction
D. R. Stinson,
Combinatorial designs with selected applications,
available from
http://cacr.math.uwaterloo.ca/~dstinson/papers/designnotes.ps
A detailed account of BIBDs, with many examples and
uptodate applications.
Format: PostScript
Contents:
 Basic definitions and properties of BIBDs
 Symmetric BIBDs
 Resolvable BIBDs
 Steiner triple systems
 Orthogonal Latin squares
 Authentication codes
 Threshold schemes
 Visual cryptography
 Group testing
 Twopoint sampling
 References
J. A. Thas and H. Van Maldeghem,
Embeddings of geometries in finite projective spaces,
available from
http://cage.rug.ac.be/~fdc/intensivecourse2/hvmjat.pdf
A timely survey of polar spaces, generalised polygons,
and partial geometries, directed towards discussing their embeddings in
projective spaces.
Format: PDF
Contents:
 Definitions
 Some important finite pointline geometries
 Embeddings of generalized quadrangles
 Embeddings of polar spaces
 Embeddings of partial geometries
 Embeddings of the flag geometries of projective planes
 Embeddings of generalized hexagons
 Polarized, flat and lax embeddings of generalized polygons
 Open cases and conjectures
 References
H. Wilf, East side, west side, available from
http://www.cis.upenn.edu/~wilf/eastwest.pdf
A set of notes accompanying a
Maple package
which can count, list, choose a random member
of ..., combinatorial objects of various types (subsets, partitions or
permutations).
Format: PDF
Contents:
 Introduction
 About programming in Maple
 Sets and subsets
 Permutations and their cycles
 Set partitions
 Integer partitions
 ... and all around the town
 The EastWest Maple package
 Program notes
Other sources for lecture notes on the Web include
Peter J. Cameron
20 May 2003