If you find a mistake, omission, etc., please let me know by e-mail.

Apropos of mistakes etc., here’s a short list of corrections to the textbook from previous editions of Math 155r.

Also, the existence of nontrivial
Steiner systems with *t* ≥ 6

The orange balls mark our current location in the course, and the current problem set.

h0.pdf: 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].

Section: Mondays 1-2 PM in Science Center 111

CA office hours:~~Mondays~~Wednesdays 8-9 in Pforzheimer Dining Hall,

except that the Apr. 27 Office Hour is moved to Apr. 26 to precede the second midterm

First Lowell House Office Hours: Tuesday Feb.9, 8:00-9:30 in the Dining Hall;

Second Lowell House Office Hours: Tuesday Feb.23, 8:00-9:30 in the Dining Hall;

Third Lowell House Office Hours:ThursdayMarch 3, 8:00-9:30 in the Dining Hall;

Fourth Lowell House Office Hours: Thursday April 14,8:30-10:30in the Dining Hall

Fifth Lowell House Office Hours: Monday April 25, 7:25-8:55 in the Dining Hall

“*faculty legislation requires all
instructors to include a statement outlining their policies regarding
collaboration on their syllabi*” —
as stated in h1.pdf: *for homework*,
“As usual in our department, you are allowed — indeed encouraged
— to collaborate on solving homework problems, but must write up your
own solutions.” *For the final project or presentation*,
work on your own even if another student has chosen the same topic.
(As with theses etc. it is still OK to ask peers to read drafts of your
paper, or see dry runs of your presentation, and make comments.)
*In all cases*, acknowledge sources as usual, including peers in your
homework group.

h1.pdf:
*Ceci n’est pas un* Math 155 syllabus.

h2.pdf:
Handout #2, containing
some basic definitions and facts about *finite fields*.

The **first midterm examination** will be **Monday, March 7
in class**. It will cover only material in problem sets 1-4 (so that
you can study your graded solution sets while preparing for it).

h3.pdf:
Handout #3: outline of a proof of the simplicity of
_{2}(F)_{n}(F)

**corrected** April 6: the PSL_{2}(F) conjugates of x↦x+c
*include* x↦x+a^{2}c, not
“are” x↦x+a^{2}c

h4.pdf:
Handout #4: The exceptional isomorphism
_{2}(**F**_{7})=GL_{3}(**F**_{2})

Here are Andries E. Brouwer’s tables of strongly regular graphs. For instance, the first table shows all parameters for v≤50 allowed by the integrality condition. Green means the graph exists, in which case the first column has "!" if it is unique up to automorphisms, and "n!" with some n>1 if the number of isomorphism classes is known to be exactly n (see the comments column: if there are no comments, look at the entry above for the complementary-graph parameters). Red means there is no such graph (and the comments indicate why not). Yellow means that existence is an open question; there is no such case for v≤50, but the next page (for 50<v≤100) already shows a few examples.

a reference for practically all the group theory
we shall need, and much more, is Joseph J. Rotman’s
*An Introduction to the Theory of Groups* (Springer 1995),
which you can view, and download (for personal use only), from
hollis.harvard.edu
on a Harvard computer.

The **second midterm examination** will be **Wednesday, April 27
in class**. It will concentrate on material from problem sets 5-8,
but 1-4 are also fair game.

The **final project** is due **Thursday, May 5 at 11PM**.
*Unlike the problem sets*, this is *not* collaborative:
you must work on your own, even if a classmate is writing on the same topic.
(You may ask friends to read drafts, but you probably prefer to ask this
of me…) *Unlike the midterms*, here you *may* (and should!)
use references — as usual cite them properly, and use your own words
in your written project. Here
is the list of sample topics that I described in class April 8.

Our April 29 meeting will be in **Room 310**,
to accommodate the meeting of the Friends of Harvard Mathematics
which will take place in 507 for much of the day. (Yes, that is during
Reading Period but I expect I'll want to make up at least one class.)

Informal lecture notes:

January 27:
Introduction: basic definitions and questions
[\D,\B are script D and B;
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;
alternative proof of Fisher using the “variance trick”
(equivalently, the

February 3 and 5:
Important examples of designs, I:
projective planes, and higher-dimensional projective spaces;
uniqueness and automorphisms of _{2}

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 λ@ --
I don't much like this but couldn't think of anything better]

Here’s my
mathoverflow
writeup of (a generalization of) the technique we introduced today for
solving Diophantine equations of the form
**Z**

February 12:
Introduction to (arcs and) ovals in square 2-designs;
a bit about intersection triangles
[\E is script E]

February 15: No class, Presidents Day

February 17:
Affine and inversive planes

February 19: No class, I was speaking at
SUnMaRC

February 22:
Introduction to strongly regular graphs

February 24:
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 26:
Moore graphs of girth 5; the “absolute bound”,
and a bit on the Krein bound

February 29:
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
Π_{2} (again) and Π_{3}

["ATLAS" = *Atlas of Finite Groups:
Maximal Subgroups and Ordinary Characters for Simple Groups*
by John Conway et al.]

March 2:
Preliminaries for the uniqueness and automorphism group of
Π_{4}: n-arc counts; simply transitive action of
_{n}(k)**P**^{n−1}(k)

March 4:
Uniqueness and number of automorphisms of Π_{4};
outer automorphism of S_{6} via permutations of a
hyperoval *O* lifted to _{4})*O**

March 7: First midterm examination

March 9:
More about the outer automorphism of S_{6}, and
_{n})

March 21:
The (5,6,12) Steiner system and its automorphism group M_{12}
via Aut(S_{6})

March 23:
The simplicity of the alternating group A_{n} (n≥5),
introduced via the determinant partition of the 168 hyperovals in
_{4}

March 25:
Interlude: recognizing S_{n} and A_{n} from their
index-n subgroups S_{n−1} and A_{n−1}.
Then, back to the Lüneburg construction:
The 4-(23,7,1) Steiner system D_{23} via hyperovals and Baer subplanes

March 28 and 30 (and 32):
Existence, uniqueness, and introduction to the automorphism group
_{24}_{3}(F_{4})_{3}(F_{4})_{3}(F_{4})_{3}(F_{4})

April 1: finish _{24}_{11} and M_{23}”: the notation H≤G
for “H is a (not necessarily normal) subgroup of G”;
theorems of
Lagrange (1771:

April 4:
Simplicity of _{12} and M_{24}

April 6: See above. For (0), the row and column operations of
basic linear algebra mean (after a bit of accounting for the determinant)
that SL_{n}(F) is generated by coordinate transvections
together with diagonal matrices and the signed permutation matrices
that correspond to simple transpositions. To reduce the latter two
to coordinate transvections, compute

[1 a] [1 0] [1 c] [ab+1 abc+a+c] [ ] [ ] [ ] = [ ] [0 1] [b 1] [0 1] [ b bc+1 ]and choose a,b,c to make ab+1 or abc+a+c zero. In part (1), consider the conjugate of x↦x+c by x↦a²x (which corresponds to diag(a,1/a)).

April 8: Description of some possible topics for the final project

April 11: Introduction to subgroups of

isomorphism between the groups

April 13: A

April 15: Determination of the finite fields

April 18: Outline of the proof of Dickson’s list of finite subgroups of (P)SL

April 20: S

April 22: The Hadamard matrix of order 12 and its automorphism group (which we’ll identify with

April 25: The Hadamard matrix H

April 27: Second midterm examination

April 29: Makeup lecture, roughly corresponding to part of these notes, with 1) more on M

p1.pdf: First problem set, exploring the Fano plane (and generalizations) and Petersen graph from the introductory handout.

The use of English words to encode combinatorial structure, as in {BUD, BYE, DOE, DRY, ORB, RUE, YOU} ≅ Π_{2}, is one of many bits of mathematics (and wordplay) that I was introduced to by the writings of the late great Martin Gardner. In page 208 ofMathematical Carnival(New York: Knopf, 1975) he introduces the following game:Each of the following words is printed on a card:HOT, HEAR, TIED, FORM, WASP, BRIM, TANK, SHIP, WOES.The nine cards are placed face up on the table. Players take turns removing a card. The first to hold three cards that bear the same letter is the winner. (The Canadian mathematician Leo Moser , who devised this game, called it “Hot.”)What familiar combinatorial structure does this set of words encode? Hint: “Hot” is the last of three games described on this page; the first is:Nine playing cards, with values from ace to nine, are face up on the table. Players take turns picking a card. The first to obtain three cards that add to 15 is the winner.(The endnotes for this chapter “16. Jam, Hot, and other games” cite Leo Moser, “The Game is Hot.”Recreational Mathematics Magazine, Vol.1, June, 1961, pages 23–24.) Another variation, using the words BET, BUG, CLOG, EACH, FRAUD, GEM, LAMB, MUTT, STILL, is here.

p2.pdf: Second problem set, mostly on square designs and intersection triangles.

p3.pdf: Third problem set: more about the special designs recently introduced, and a bit of inclusion-exclusion

p4.pdf:
Fourth problem set: Strongly regular graphs I

[the TeX macro I used to center the bar of
*G*
more-or-less correctly is `\def\Gbar{\,\overline{\!G}}` .]

p5.pdf:
Uniqueness of Π_{5} 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

For problem 3, you may assume that PSL_{2}(F_{q})
is simple for q>3, and thus that PGL_{2}(F_{q})
has no normal subgroups other than itself, {1}, and (when q is odd)
PSL_{2}(F_{q}). We shall prove these results soon
(they are also in Rotman).

p8.pdf (final problem set):

More on Π_{2} structures and the identification of
A_{8} with GL_{4}(F_{2});

Finite subgroups of PSL_{2}(F) and related matters