Symmetry
An Introduction To Group Theory
Survey!
Now that the course has reached its end, we'd like to get some
feedback on the course. Please see the survey (web or pdf) and email responses to Peter.
Thank you for participating in the course!
Contest Solutions!
The contest problems (PDF) now have solutions (PDF) available.
Contest champions were be announced Tuesday, January
14th: Vishu won for most challenging (solved) problem, and Anton won
for most / best solutions.
Please email complaints to
Peter.
Week Thirteen: Even More Adventures With Rubik's Cube
Today we spent some time discussing the cube. In particular, we
figured out how to tell the order of a move: we looked at UR
and found it had order 105 without (mostly) repeating UR 105
times. There was a brief discussion of how to figure out what the
largest possible order of a cube move is. (See the homework for some
comments on this.)
We also discussed the center of the cube group; that is, the set of
moves that commute with every other move. (In fact, it is sufficient
to show that a move commutes with the six quarter turns U,
D, F, B, R, and L, as these are the
generators of the cube group.) We discussed (but did not rigorously
prove) the two elements of the center: the identity 1 and the
superflip (the move that flips each edge and leaves the corners
untouched).
We also briefly discussed how to make new cube moves: from
conjugates XYX-1 and commutators
[X,Y] = XYX-1Y-1. Both
examples were from the cube solution guide from Week Eleven.
Today's discussion was based in large part on
Frey and
Singmaster and Prof. Joyner's lecture
notes. (Both give equations for the superflip. The one in class
was from Frey and Singmaster, so check out Joyner's nice lecture notes.)
Homework Problems: web or pdf (pdf recommended)
Week Twelve: Rubik's Cube: The Group
and Some Subgroups
More adventures with the Rubik's cube. Today we deduced the order of
the Rubik's cube group (and it's big!). From there we looked at some
simpler subgroups: the slice group (from which we made
checkerboards and spots), the squares group of 180 degree turns
only, the two-squares group (isomorphic to D6), and
the square-slices group (isomorphic to
C2×C2×C2).
Much of today's discussion was based on Frey and Singmaster (see the
bibliography).
Homework Problems: web or pdf (pdf recommended)
Week Eleven: Cubes and Braids
Today was the long-awaited introduction to the Rubik's Cube. I've
written out a solution with pictures in color or
black and white (both pdf, updated January 8th).
Due to the dearth of cubes, we collectively got most of the cube done
and (I hope) familiarized everyone with the technique. (Note to those
who weren't in class: the pictures in the solution guide uniformly
show the front, right, and top sides of the cube. That is, the
F, R, and U sides.)
After some late pizza, we discussed braids as a continuation of our
discussion of knots last week. See the homework for some discussion
of braids and the centre (note spelling!) of the braid group.
Homework Problems: web or pdf (pdf recommended)
Week Ten: ``Basic Units'' and Basic Knots
We began the day with chapter 1 of Farmer on what he calls
basic units. This was really an excuse to discuss what happens
when you identify all elements that are equivalent under the symmetry
group: we talked about tori (the plural of torus) and Klein bottles.
After some early pizza, we moved on to knots. I passed out some
copies of hand-drawn knots, unknots, and a link (see Peter if you
missed them). We discussed the knot group, and using the Wirtinger
presentation of this group as a way to find some obscure presentations
(see also the contest!). We figured out the knot group of the unknot
(it's Z, the integers) and wrote down a presentation of one
particular unknot (it was Z too, as it should be).
Finally, we tried to find knots with few crossings, written with the
fewest possible crossings. We decided that: the unknot was the only
knot with no crossings, there are no knots with only one or two
crossings (when simplified), and only two with three crossings (the
right and left trefoil knots). We decided there were two knots with
four crossings (both variants of the figure 8 knot). These two aren't
really different, however: one is simply the other turned upside
down. See the bibliography
for some pages with pretty pictures and catalogs of knots given by
their minimal crossing number. This part of the lecture was very
sketchily based on chapter 3 of Farmer and Stanford.
Homework Problems: web or pdf (pdf recommended)
Week Nine: Homomorphisms and Infinity
After dancing around the concept of isomorphismic groups for a few
weeks, we finally made a formal definition of isomorphism as a
one-to-one (injective) and onto (surjective) homomorphism (or
bijective homomorphism). This led us to the ideas of kernel, image,
and preimage that are discussed more on this week's homework.
One application of one-to-one functions is that of comparing
cardinality, or size, of sets. We showed that the following sets have
the same cardinality: the positive integers, the integers, and the
rationals (fractions). On the other hand, the reals have a strictly
greater cardinality than these other sets. The continuum
hypothesis states that there is no set with cardinality strictly
between that of the integers and the reals. (This is not one
of the Clay Mathematical
Institute's million-dollar Millenium
Prizes as I suggested.)
See the bibliography
for more links on these topics.
Finally, I promised a link to information on the Putnam Competition,
sponsored by the Mathematical Association
of America. Here
is a page with recent exams and solutions.
Finally, don't forget the contest! (See homework 8, below.) The
deadline is now Friday, December 6th at noon.
Homework Problems: web or pdf (pdf recommended)
Week Eight: More On Finite Groups
We reviewed our knowledge of finite groups (see today's handout) to consider the
following question: how many different (that is, non-isomorphic)
groups of a given order are there? This is, of course, a fairly
complicated question, but we managed to make a little bit of headway
and prove (or deduce) some partial results. There are, of course,
many web sites devoted to this; see the bibliography for some links.
This week's homework is meant to be (mostly) approachable and includes
an arbitrarily judged contest (see problem 3). Entries due in class
next week.
Homework Problems: web or pdf (pdf recommended)
Week Seven: Wallpaper and Frieze Patterns
We mostly covered chapters 4 and 5 of
Farmer today. That is,
we discussed the seven frieze groups (or groups of symmetries of
infinite strips with (discrete) translation symmetries) and then moved
on to the 17 groups of wallpaper symmetries. There are lots of pretty
pictures on the web; see the bibliography.
We did begin the day looking at some of last week's homework and
applying it to the RSA algorithm. Again, there's plenty of
information about this on the web, linked through the
bibliography.
There are no homework problems this week. Don't forget
to buy a Rubik's cube!
Week Six: More on the Symmetric Group,
the Euler Phi Function, and Equivalence Classes
We began today looking again at cycles and found that they come in
even and odd varieties, and the set of even permutations is a subgroup
of the full symmetric group. After some examples (mostly about
shuffling cards) we returned the groups of multiplication modulo
n we examined last time. It turns out the order of these
groups is the famous Euler phi function (or totient function, see the
homework). After some heavy group theory, we proved Wilson's
Theorem, which says that if p is an odd prime, then
(p-1)! is equivalent to -1 modulo p.
We ended the day with a brief discussion of equivalence relations (see
homework five) and conjugacy classes (see homework six).
Homework Problems: web or pdf (pdf recommended)
Week Five: Subgroups and Some New Groups
This week we returned to subgroups, discussing how subgroups can be
obtained by breaking symmetries of finite figures. We introduced
modular arithmetic and this produced a number of accessible groups
(see the homework for the proof of a conjecture from class). Through
the symmetries of the tetrahedron, we introduced cycle notation and
the symmetric group. Much of this week (except for the tetrahedron)
was adapted from Farmer,
chapter 6.
Homework Problems: web or pdf (pdf recommended)
Week Four: Tilings Of The Plane
This week we discussed some basic tilings of the plane. We uncovered
the eleven Archimedian (or uniform)
tilings as well as a few two-uniform tilings. A tiling is
k-uniform if there are k distinct vertices after
identifying any two vertices that can be moved to each other by a
rigid motion of the plane that preserves the entire tiling. (A
uniform tiling is simply a one-uniform tiling.) The standard book on
tiling is by Grunbaum and Shephard.
See the bibliography for some
links to some cool pages (and computer programs) on tilings.
Homework Problems: web or pdf (updated late Tuesday, October 29th)
Week Three: Symmetries of the Platonic Solids
This week we discussed the symmetries of the five Platonic solids: the
tetrahedron, the cube, the octahedron, the dodecahedron, and the
icosahedron. We discussed the idea of duality, and showed that the
tetrahedron is self-dual, and the others are pairwise dual (the cube
and octahedron are each duals of the other, as are the dodecahedron
and icosahedron). We conjectured (prove this!) that the order of the
tetrahedral group (the symmetry group for the tetrahedron) is 12,
and that the order of any of the symmetry groups is twice the number
of edges (see the homework
exercises for this and other in-class conjectures).
Mathematically, we described more examples of groups, defined the
order of both a group and an element, defined subgroups (with
examples), and defined abelian groups. See the homework exercises for definitions
and some applications.
Homework Problems: web or pdf
Week Two: Symmetries of Finite Figures
This week we discussed the symmetries of finite figures, most notably
the regular n-sided polygons. This led us to the definition of
a group, and we discussed several examples (and non-examples). Much
of this (although notably not the explicit discussion of groups, which
appears later in Farmer) was more or less based on Farmer, chapter 3.
Homework Problems: web or pdf
Week One: Rigid Motions of the
Plane
This week we discussed the four kinds of rigid motions of the
plane: the translations, the rotations, the
reflections, and the glide reflections. Although we did
not prove this, it seemed to be the case that any combination of two
rigid motions resulted in a single rigid motion (of one of our four
types). Our discussion was more or less based on Farmer, chapter 2.
Homework Problems: web or pdf
Annotated Bibliography