h1.pdf:
Ceci n'est pas un Math 155 syllabus.
If you find a mistake, omission, etc., please let me know by e-mail.
The orange balls mark our current location in the course, and the current problem set.
h0.ps:
introductory handout, showing different views of the
projective plane of order 2 (a.k.a. Fano plane)
and Petersen Graph [see also the background pattern for this page]
h1.pdf:
Ceci n'est pas un Math 155 syllabus.
h2.pdf:
Handout #2, containing
some basic definitions and facts about finite fields
h3.pdf:
Handout #3, containing a sketch (to be filled-in
in class) of the existence and uniqueness of
the Moore graph of degree 7, a.k.a. the
Hoffman-Singleton graph. Can you deduce
the size of the automorphism group of this graph?
h4.pdf:
Handout #4: outline of a proof of the simplicity of
corrected and expanded Apr.2 (n≥3 page)
h5.pdf:
The excpetional isomorphism
A list of sample topics
for your final projects
Informal lecture notes:
January 27:
Introduction: basic definitions and questions
[\D,\B are script D and B; \lam=lambda=λ;
Bin(n,k) = binomial coefficient
January 29:
Duality and the incidence matrix of a design; Fisher's theorem
[\T = transpose]
February 1:
Square designs continued: theorem of Bruck-Ryser and Chowla;
dualities and polarities; alternative proof of Fisher using
the “variance trick” (equivalently, the Cauchy-Schwarz
inequality), which generalizes to an inequality on arbitrary 2-designs
Errors/typos in the textbook: on page 6,
in the first display both instances of fancy script \B
should be plain B (it's a single block, not the collection of
all blocks); and on page 7, part (i) of Theorem 1.21 should have
n=k-λ a square, not k.
February 3 (and 5):
Important examples of designs, I:
projective planes, and higher-dimensional projective spaces;
uniqueness and automorphisms of
February 5 (and 8):
Important examples of designs, II:
“Hadamard 2-designs” (square
February 10:
New designs from old: complement, Hadamard 3-designs, derived designs
[@ is an \overline (a.k.a. vinculum) for design complements, so
\D@ is the design complementary to \D, and likewise \B@ and \lam@ --
I don't much like this but couldn't think of anything better]
Errors/typos in the textbook: on page 11, the reference to
(1.7) preceding the statement of Proposition 1.33 should be to (1.8);
Proposition 1.34 should really be Hughes' 1961 result that n is in
{2,4,10}, because we do not yet know that n=4 works, and will not prove
that n=10 doesn't occur.
February 12:
Introduction to (arcs and) ovals in square 2-designs;
a bit about intersection triangles
[\E is script E; == is the congruence symbol ≡;
\nu is the Greek letter ν]
Yucko in the textbook: in the proof of Proposition 1.48
on page 18, please change ni to Ni
to avoid the ugly nn.
February 17:
Affine and inversive planes
February 19:
Introduction to strongly regular graphs
[\mu is the Greek letter μ]
February 22:
The adjacency matrix of a graph (not necessarily regular), and the
integrality condition on the parameters of a strongly regular graph
[\j is a boldface j, denoting an
February 24:
Moore graphs of girth 5; the “absolute bound”,
and a bit on the Krein bound
February 26 (and March 1):
Overview of the second part of the course, where groups will
play a more central role; introduction of some of our techniques via
uniqueness and automorphism group of
Π2 (again) and Π3
[=~= is the congruence symbol ≅]
corrected Feb.26 after class to include the exceptional
isomorphisms involving the not-quite-simple groups of order 24, 120, 336
and fix the proofs of the uniqueness of Π2 and Π3
March 1:
Preliminaries for the uniqueness and automorphism group of
Π4: n-arc counts; simply transitive action of
March 3:
Uniqueness and number of automorphisms of Π4;
outer automorphism of S6 via permutations of a
hyperoval O lifted to
March 5:
More about the outer automorphism of S6, and
March 8:
First midterm examination
March 10:
The (5,6,12) Steiner system and its automorphism group M12
via Aut(S6)
Error/typo in the textbook: The intersection triangle on
page 87 for this Steiner system (Table 6.1) should have 3,2,3 in the
middle of the bottom row, not 2,3,2
[March 12: existence, uniqueness, and automorphism count of the
Moore graph of degree 7, following
the third handout above]
March 22:
The simplicity of the alternating group An (n≥5),
introduced via the determinant partition of the 168 hyperovals in
March 24:
Existence, uniqueness, and introduction to the automorphism groups
of the
corrected Mar.26 after class to fix the intersection triangle
Typo in the textbook the next-to-last line
of page 21 should have “in accordance with (1.52)”, not (1.42).
(noted by N.Kaplan, who also lists a few apparent
typos in chapters 4 and 5,
which we won't cover in this class)
March 26 (and 29):
Existence, uniqueness, and introduction to the automorphism group
March 29: Simplicity of M11 and M23, following
Robin J. Chapman's “An elementary proof of the simplicity of
the Mathieu groups
March 31:
Simplicity of
April 2: see above
April 5: Introduction to subgroups of
April 7:
A4, S4, A5,
and the other finite triangle groups via Cayley graphs
April 9:
Determination of the finite fields Fq
for which
April 12:
Outline of the proof of Dickson's list of finite subgroups of
(P)SL2(K) for any field K. We follow Suzuki's
exposition, concentrating on the subgroups whose order is
not a multiple of the characteristic, which is what we need for
Galois' theorem on the smallest permutation representation of
April 14:
S4 and A5 twins in PSL2(F);
the twin
April 16:
The Hadamard matrix of order 12 and its automorphism group
(which we'll identify with
April 19:
The Hadamard matrix H12 and the Mathieu group M12
(and the action of
April 21:
More on M12, the (5,6,12) Steiner system, and the
affine
corrected Apr.21 after class to clarify the argument for
getting from O to all 9+9 translates of O and O'
April 23:
A bit on the Golay [24,12,8] code; the 2576 “umbral dodecads”,
and M12.2 inside M24
April 26: A bit more about the (3,4,10) design;
overview of the maximal subgroups of M24
April 28:
Second midterm examination
p1.pdf:
First problem set, exploring the Fano plane (and generalizations)
and Petersen graph from the introductory handout.
p2.pdf:
Second problem set, mostly on square designs and intersection triangles.
Problems 6 and 7 postponed till next week.
Problem 1 corrected [L.Mocz]: the first display should sum
p3.pdf:
Third problem set: more about the special designs recently introduced,
and a bit of inclusion-exclusion
Problems 4 and 5 corrected [Z.Abel and C.Anderson]:
for the last part of Problem 4, the field F must
be perfect (which is automatic for a finite field but not in general,
see the new footnote); for Problem 5, the condition
As was the case last week, and if it ever happens again,
any problems affected by an error that requires correction
can be handed in without penalty at the class meeting after
the original due date if the error was not announced by the
previous meeting (so in this case problems 4 and 5 are due
Feb.19 instead of Feb.17).
p4.pdf:
Fourth problem set: spherical 3-designs; regular graphs, cont'd
Corrected: of course it's February 2010, not 2009...
p5.pdf:
Uniqueness of Π5 via ovals, synthemes, and totals
[“syntheme” = one of the 15 partitions of a six-element set
into three pairs; “total” = one of the 6 collections of
synthemes that cover each of 15 pairs once; the text calls these
“1-factors” and “1-factorizations” respectively
(page 81).]
p6.pdf: Some finite group theory
p7.pdf: The Tutte 8-cage; transvections;
the 4-dimensional representation of
Problem 3 corrected [C.Anderson]: in part (i), the condition
p8.pdf: