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Group Theory & Rubik's Cube
Jim Mahoney
(mahoney@marlboro.edu)
Contents
General Info
 Time
 M,Th 1:30
 Place
 SciBldg 217
 Credits
 2 or 3


Group theory is the study of the algebra of transformations and symmetry.
While that sounds a bit esoteric (and it certainly can be), what
it means is that it looks at the ways you can turn, rotate,
or stretch one pattern or dohickey back onto itself  which is something
that puzzles like the Rubik's Cube and pictures like the ones Escher drew
have in common.
This course is an introduction to group theory using various puzzles
as examples to make the subject more accessible and concrete.
The level will depend on who shows up: at one extreme, some of us
can taste the edges of a very beautiful piece of mathematics while
learning to solve the Rubik's cube, while at the other extreme, some
of us may delve into some deep mathematical proofs. We'll see where
we want to head, and how far, depending on your backgrounds and interests.
After our initial discussions, it now seems that folks
doing the 2 credit version need only come on Thursdays,
when we will focus on the puzzles and general ideas,
while those who want to see more of the proofs and
deeper mathematics should come Mondays as well, for
the 3 credit version.
Each of you will do a term project, usually the study of
a specific puzzle using the group theory methods we develop in
class. More on this as the term goes on..
I got started on all this back in my undergraduate days when I
did my bachelor's thesis on the Rubik's Cube, back when it was hot
in the early '80's. And these days group theory is also a central
part all the fundamental physics theories, too  so it isn't only
useful for toys. Though of course toys are fun...
Notes
Assignments (thursdays, for 2 cr folks and 3 cr folks)
 for Thurs Jan 20
 read "What is GROUP THEORY"
 for Thurs Jan 27
 read "Examples"
 Find the multiplication table for D3, the symmetries (including) "flipping over" of a triangle.
 Must it be true that if for elements of a group, if b=a^{1}, then a=b^{1}? Give a counterexample, or argue why this must be true. Which group properties does this depend on?
 Are the integers a group? Explain.
 Rational numbers are all those that can be written as p/q where p,q are integers.
Do the set of all rational numbers and the operation of muliplication form a group?
Explain.
 for Thurs Feb 3
 Find the elements and multiplication table for the symmetry
group of a solid 3D cube.
 Are any of the groups D4, S4, A4, C4, S5, A5, D5, C5 isometric?
 for Thurs Feb 10
 Find the commutator subgroup of the solid cube group.
 Do R, F generate the cube group? R,F,U? R???
 What is the cycle length of the Rubik's commutator FRF'R' ? FR'F'R ?
 for Thurs Feb 17
 Let an element of the permutation group S5 be A = (1 2 3 4 5  2 3 1 5 4).
Express this in "cycle" notation. If A^n=I, what is n? Is A odd or even?
 Let another element of S5 be B = (1 2 3 4 5  2 3 4 1 5). Again,
find the cycle representation, m such that B^m=I, and whether its odd or even.
 Using A and B from the last two problems, find A*B, B*A, [A,B].
 Find an element "a" that is conjugate to A and an element "b"
which are conjugate to B. (Both elements of S5.)
 How many distinct Rubik positions are there 2 moves (quarter turn) from solved?
3 moves? 4?
 Start trying to think about how to solve the corners on a Rubik's Cube.
Do the commutators of R,L give interesting "operators"? (yes).
Can you use 3cycles? How are you thinking about what you're doing?
 for Thurs Feb 24
 Continue thinking about how to solve the Rubik's Cube:
can you find a sequence of move that cycles three corners?
Should that be an even or an odd number of moves? (Is it an even
or odd permutation?)
One answer: (FRF'R')B(RFR'F')B'  what is going on in terms
of permutation cycles?
 Read Dog School notes, chap 1214, and be ready to discuss.
 for Thurs Mar 2
 Again, continue thinking about how to solve the Rubik's cube
based on our class discussion. How many of the algorithms
from the rubik10.txt notes can you find?
 How many positions are there in the subgroup generated by
only the R and F moves? (the "2face" subgroup)
 See if you can come up with a stepbystep
written algorithm for getting back
to the solved position from any element, i.e.
First step: use the () operator which does () to get () pieces solved.
Second step: use the ...
 These kinds of solution techniques are
called "nested subgroups" method of solution. Why?
 Thurs Mar 9
 How many distinct groups are there of order 8? Name and/or describe them.
(Hint: Lagrange's Theorem provides some assistance.)
 Finish the RF algorithms if you haven't already.
 Submit an initial proposal for a final project.
 Due after break: describe a "solution" to the 3x3 Rubik's Cube.
 Thurs Mar 31
 Describe a method for "solving" the 3x3 Rubik's Cube.
 Stuff you don't have to hand in:
 Read about "Tilings and Tesselations"; see Resources list below
 Play with Kali
 Thurs Apr 6
 Peruse and think about the group nature of the 17 wallpaper groups,
listed here. Pick one we didn't discuss in class,
and describe a few entries in the group multiplication table.
Find two group elements that don't commute, and calculate their
commutator.
 Pick one of the "star patterns" in the
Wallpaper Gallery, and explain which of the 17
wallpaper groups it belongs to and why.
 Thurs Apr 13
 Peruse the change ringing references
 Write out a few more lines in the "Stedman" principle changes,
described the campanology_math_refs document.
(You can find it by searching for "stedman", but you'll need to read enough to figure out what the heck all those symbols mean...)
 Thurs Apr 20
 In class: 3D space groups and crystals. References below.
 Work on your projects.
Extra Assignments (occasional; for monday 3 cr folks)
 for Mon Feb 7 (originally for 1/31; postponed a week)
 Check out first few chapters in yellow
Armstrong's Groups and Symmetry.
 Do 2.5, 2.7, 3.3 in that text. Be clear about definitions and
what you need to prove or show what.
2.5: Prove that (a) an isometry is a bijection and
(b) the set of all isometric plane transformations is a group.
2.7: If {x,y} are members of group G, prove that there exists
w,z such that (i) wx=y, zw=y and (2) w,z are unique.
3.3: Show that for an integer n and complex c,
the set {c  c^{n}=1} forms a group. (What group
that you already know are these isomorphic to?)
 for Mon Feb 14
 Read the Housekeeping chapter in the Dog School notes
 Read chapters 0 and 1 in Joyner's notes.
 Use Venn diagrams to verify the DeMorgan laws (Joyner, chap 0)
 Show that group isomorphism is an equivalence relation (Joyner, chap 1)
 Show that any equivalence relation implies a partition into disjoint sets.
 Show that "conjugacy" (see buzzword definitions) is an equivalence relation.
 for Mon Feb 21
 Prove that the "solid cube group" is isomorphic to S4. (See Armstrong, chap 8)
 Practice some matrix operations if you need to: multiplication, determinant, inverse,
as described in class and/or in any standard calculuslevel text.
 Armstrong, 9.12: Proove that these four matrices form a subgroup of SO3, and
find the corresponding rotations.
 Armstrong 9.8: Show that these two matrices represent rotations, and
find the angle and axis for each:
2/3  1/3  2/3 
2/3  2/3  1/3 
1/3  2/3  2/3 

1/sqrt(2)  1/sqrt(3)  1/sqrt(6) 
1/sqrt(2)  1/sqrt(3)  1/sqrt(6) 
0  1/sqrt(3)  2/sqrt(6) 

 for Mon Feb 28
 Prove or disprove: Any element "a" from a group "G" is conjugate
to it's inverse "a'", i.e. there exists "b" such that
b a b' = a' . (Hint: think Abelian.)
 Show that C_{n,m1} + C_{n,m} = C_{n+1,m}, i.e.
that Pascal's Triangle works. As usual, C_{n,m} = n!/(m! (nm)!) .
 Find a 3x3 matrix corresponding to one of the 120 degree
corner rotations of the solid cube. Hint: use the
90 facecentered ones we did in class as generators
to get them. For example, a rotation by theta about the Z axis is
is
[ cos(theta) sin(theta) 0 ]
[ sin(theta) cos(theta) 0 ]
[ 0 0 1 ]
Just put theta=90 to get one of the 90 facecentered moves.
Also, don't forget the righttoleft vs lefttoright multiplication issue.
 for Mon Mar 6
 (If you know determinants). Show that the determinant of a matrix
with an odd number of columns swapped in the identify matrix is 1.
 If X and Y are boosts in the X and Y directions, find the matrix
for X Y X' Y' and show that it's a rotation. (Lorentz Group)
 10.11 in Armstrong: Show that D_2n is isomorpnic to D_n X Z_2
when n is odd. (What happens when n is even?)
 for Mon Mar 13
 Read Joyner chap 5, 6, 9, on homomorphisms and quotient groups.
Or Armstrong, 15 and 16
Have a good spring break.
 Mon Mar 27
 In class: factor decomposisition of cube group?
Are there any nonAbelian simple groups?
 Mon Apr 3
 Show that A5 is a simple group. (It's the smallest, actually.)
 Look up "simple group" in online MathWorld,
surf through some of the links there (monster group, Mathieu,...)
 Surf "atlas of group representations"
 Mon Apr 10
 Log into the "GAP" program on big, as describe in class. Read a bit of the
online manual (see the links below) and use it to do at least one of the
assignments we've tried earlier in the term. Can you define the solid cube group
in GAP? Try putting the numbers 1 through 6 on the faces, and then finding
a few generators.
Buzzwords, Notations, Definitions, and Theorems
A collection of various technical geeky words and
their (fairly) precise definitions that many mathemeticians
have at the tip of their tongue, but you and I may
want to have a place to look up. (Expect this table will
grow as the class progress. Also expect errors, which you
should call to Jim's attention.)
1to1 
A map from {x}>{y} in which every x is sent to a different y,
i.e. x not equal y implies f(x) not equal f(y). This does not
imply that the map must completely cover {y}. (See "onto", "bijection".)

Abelian 
A group G is called abelian if a*b=b*a for all a,b in G.
(See "boring.")

Automorphism 
An automorphism is an ismomorphic map of a group G onto itself.
The set of isomorphisms of a given group G is also a group,
called Aut(G).

Bijection 
A bijection is a map which is both 1to1 and onto.
Such maps take each element of {x} to a unique {y}, and
each {y} is the result of a unique {x}; therefore, bijections
are invertible maps and sometimes written {x} <> {y} .

boring 
A group G is called "boring" if it isn't particularly related to
any interesting puzzles, physics, or profound cool group stuff.
At least in Jim's opinion. (see "Abelian".)

Campanology 
The art and science of "change ringing", ringing N
church (or hand) bells in many of their possible
N permutations in a systematic way.
Folks who did this
understood quite a bit about permutations long before
the mathematics of "group theory" existed; therefore,
you could say it was the precursor of all this stuff.

Conjugate 
Two group elements a and b are conjugate to each other
if there exists an element c such that a = c b c^{1} .
The set of all {x} conjugate to a given element y is called
y's "conjugacy class." It turns out that these conjugacy
classes partition each group into disjoint subsets; each
group element belongs to a single conjugacy class.

Cayley's Theorem (All groups are permutations.) 
Every group G is is isomorphic to a subgroup of S_{G}, the
group of permutations of the elements of G.
If G is finite and has order n, then G is isomorphic to a subgroup
of S_{n}.

Countable 
A set X is countable iff there exists a 1to1 map from the positive
integers to X, {1,2,3,...} <> {X}.

Commute 
Two group elements a,b are said to commute if a*b=b*a.
(See "Abelian.")

Commutator 
The commutator [a,b] of two elements is a*b*a^{1}*b^{1}.

Commutator Subgroup 
The set of all {x} such that x = [a,b] for some a,b in a group G
is called that group's commutator subgroup. The order of this
subgroup is a measure of how abelianlike G is.

Coset 
Given a group G with a subgroup H={h1,h2,...}, the "left coset"
of H corresponding to an element x of G is defined as the set
{ x h1 , x h2 , x h3, ... }.
"x is in the same coset as y" defines an equivalence relation
between x and y, and thus partitions G into order(H) disjoint
sets. Showing that each of these cosets has the same number
of elements leads to a proof of Lagrange's Theorem.

cubelet 
One of the smaller solid cubes which together make up
the Rubik's Cube puzzle. Each of the corners, edges, and faces
in the 3x3x3 Rubik's Cube is a "cubelet."

Cyclic 
The cyclic groups ( Cn ) are those isomorphic to the integers
{0,1,2,3,...,(n1)} under addition mod n.

Determinant 
The determinant of an NxN square matrix is the scalar value of
the Ndimensional "volume" spanned by the column vectors of
the matrix. In particular, if the determinant is zero then
the matrix has no inverse, is is not particularly interesting.
For a 2x2 matrix (a,b; c,d) the determinant is a*db*c. For
larger matrices the formulas get trickier; check any linear
algebra or calculus text.

Examples 
Some specific named groups discussed in class and (brief) definitions:
 C_{n} : cyclic group of n elements; order = n
 S_{n} : permuntations of n objects; order = n!
 A_{n} : even permunations of n objects; order = (n!)/2
 D_{n} : rotations and flips of plane ngon; order = 2n
 Euclidian group : isometries (distances stay same) of plane; uncountable
 Symmetries of regular solids: cube (order 24), tetrahedron, etc.
 Rubik's Cube: order = 12! 2^{12} 8! 3^{8} / 12 = 43252003274489856000.
(This is for face centers held fixed in space and in any orientation,
not counting positions reached by dissesembling cube.)
 Matrix groups
 GLn(R) : "general linear group": n x n invertable matrices with real entries
 On : n x n orthogonal real matrices (subgroup of GLn)
 SOn : n x n orthogonal, determinant=1 (special) matrices (subgroup of On)
 Un : n x n unitary (complex analogue to orthogonal) matrices (subroup of GLn(C))
 SUn : n x n unitary, determinant=1 matrices

Equivalence relation 
An equivalence relation on a set S is a set of pairs
of elements (s1,s2) with the following
properties:
 a~a for all a in S,
 a~b imples b~a,
 if a~b and b~c , then a~c .
Any such relation breaks S into equivalence classes
(sets of elements equivalent to each other), and these
classes partition S into disjoin subsets. Typically
we write s1~s2 using one of the various "=" symbols.

Function 
A function F(x)=y is a map between two sets, {x}>{y}. (See Map.)

Generators 
The generators of a group G are elements in a subset H of G = {h1,h2,h3...}
such that any element of G may be reached by a some sequence
x1*x2*x3*x4*... where all the x's are members of H.
The set H is not usually unique.
(Any subset of elements {a,b,c,...} of G similarly generates
a group which must be a subgroup of G.)
The order of the smallest possible
such set H may be though of as a characterstic "dimension"
of the group, analogous to the 1,2,3 dimensions of a point, line,
plane in Euclidean space, i.e. as the number of independent
"directions" which extend outwards from the origin (identify).

Group 
A set G = {a, b, c,...} and a binary operation *
with the following properties:
(i) closure: For any a, b in G, a * b =
is in the set.
(ii) associativity: For all a, b, c in G,
(a * b )* c = a * ( b * c ) .
(iii) identify: There exists an element I such that
I * a = a for every a in G.
(iv) inverse: For every element a there exists an
a^{1} such that
a * a^{1} = I.

Homomophism 
A homomorphism from G into H is a map f which perserves the group operation,
i.e. for all g1, g2 in G, f(g1 g2) = f(g1) f(g2).
See kernel.

Isomorphic 
Two groups G and H are isomorphic if and only if there exists a 1to1 map
between them which preserves the group multiplication table. In other
words, if g1 and g2 are members of G, and h1=f(g1), h2=f(g2) are the
corresponding members of H under the 1to1 map f, then f(g1*g2)=f(g1)*f(g2).
Intuitively, isomorphic groups are essentially the same for all practical
purposes.

Kernel 
The kernel of a homomorphism G>H is the set of elements of G which
are mapped to the identify of H.
The kernel is always a normal subgroup of G, and its cosets form
a quotient group G/(kernel) which is isomophic to H.
See quotient group.

Lagrange's Theorem 
The order of a subgroup H of a group G divides the order of G.
(See "coset" for an outline of a proof.)

Map 
A relation between two sets X={a,b,c,...} and Y={A,B,C,...} that
given any element in X specifies an element of Y. Often written
as X>Y. The relation may be given explictly or (more often) by
some kind of rule. Every element of X _must_ be mapped to something
in Y. The converse is not necessarily true; there may be "untouched"
elements in Y. X and Y may be the same.
(See Function, 1to1). Example: X={1,2,3}, Y={1,2,3,4},
x>1 for any x in X. (As a function f, f(1)=1, f(2)=1, f(3)=1.)

Matrices 
There are several ways to define these depending on how picky you
want to get. Technically, matrices are linear maps in an Ndimensional
vector space, which can be written as a table of numbers in a given
basis for the vector space. Practically speaking, the matrix is usually just thought of
as that NxN table of numbers. For example, the 3x3 "identity" matrix looks like this:
Matrices are worthy of a whole course unto themselves (called
linear algebra), in which you learn about their inverses, determinants,
eigenvalues, eigenvectors, diagonalization, and a whole lot of
other multisyllabic words that we won't get into here. But
the easy parts are so common and so useful in group theory that
we will make some use of them. Most calculuslevel or even
precalculus level textbooks discuss some of the operations
you can perform with matrices, and we'll go over some of
the basics of how they work and how they look from the
perspective of group theory in class. See, for example,
MatrixMultiplication.html or more examples. (I may have a more detailed online tutorial later.)
Here's a javascript 3x3 matrix calculator.
Also see determinant.

Onto 
A map f:{x}>{y} such that for each element y there exists
an x such that f(x)=y; i.e. the map touches every part of {y}.
(See "1to1", "bijection".)

Orbit 
Given an element x of a group G, the orbit of x is the
set of all elements of G which are generated by x,
i.e. {x, x^{2}, x^{3}, ... }.
For any element x, the orbit of x is a subgroup
of G isorphic to C_{N}, the cyclic group of N elements,
where x^{N}=I.

Order 
Generally speaking, "how many." More specifically,
the order of a group (or subgroup) is how many elements
there are in that group (or subgroup). By the order of an element
of a group we usually mean how many elements there are in its orbit,
i.e. the order of an element x is the smallest positive integer N
such that x^{N}=I.

Normal 
A subgroup J of a group G is "normal" if any of these three equivalent
conditions are met:
 J is made up of whole conjugacy classes, or
 g J g^{1} = J for all g in G, or
 the left and right cosets of J are the same, xJ=Jx for all x in G.
See homomorphism, simple, kernel

Quotient group 
Given a group G and a normal subgroup J, the set of cosets of J
form a group G/J of order ord(G)/ord(J) whose group operation is
given by
(xJ)(yJ)=(xyJ)
where each () represents one of the cosets. This is also called
"G mod J".

Representation 
A representation of a group G is a set of matrices M which
are homomorphic to the group. In other words, there must
exists a map f:G>M such that f(g1 g2) = f(g1) * f(g2)
where "*" here refers to the usual matrix multiplication.
Representations of groups is a whole branch of group theory
unto itself.

Rubik's Cube 
A grouptheory permutation puzzle made up of a 3dimensional array
of smaller "cubies" ( N^{3} of them, where N=2,3,4,...)
with colored faces. See the java applet at the top of this web page.

Scalar 
A single real or complex numeric value. (See "Vector", "Matrix".)

Semidirect product 
If a group G has a normal subgroup N, and thus can be
factored as G/N = M, then we also say that G is the
"semidirect" product of N and M, G = N x M.

Simple 
A simple group is one which has only two normal subgroups: the identity
element and the entire group. Simple groups cannot be factored,
and so are analogous to prime numbers.
(hard) Question: what is the smallest nonabelian simple group?
Answer: A5. (This fact is directly related to one of the major
math results of the last century, namely that you can solve
the general 4th order polynomial, but not the general 5th order one.)

Set 
A collection of elements {a, b, c, d, ... } of any kind. May be empty.
Size may be zero (null set, {}), finite (example: {1,2,3}) or infinite
(example: {integers}).

Subgroup 
A subset of a group which is also a group.

Vector 
Technically, a member of a linear vector space with certain kinds of addition properties
which can be written as a column of numbers given a specific basis for the vector space.
Practically speaking, we usually just imagine the vector as a column (or row) of numbers,
A = (1,0,0). (See "Scalar", "Matrix")

Resources
 mathworld  online math encyclopedia/dictionary
 Books/printed
 Winning Ways for your Mathematical Plays (on reserve)
 An Intro to Matrices, Sets, and Groups for Science Students (reserve?)
 others to come
 Groups and Rubik's Cube
 Wallpaper and other 2dimensional symmetries
 Crytals and 3d symmetries
 other Group info
 Lie Groups
 Campanology (Change Ringing)
 Rubik's Cube
 Other math puzzles and ideas for projects
 Rubik Group
 Physics
 Organizations
Topics
Here is a list of possible topics and buzzwords
as of the start of the term.
It is not at all the order we will study things,
and is at best wildly innaccurate as a list
of what we'll actually do.
But we'll definitely do some of these things,
and this will at least give an idea of where we're
headed.
Background
* proofs and mathematics
* general idea of an "algebra"
* matrices and linear algebra
Group Theory
* definition:
 elements and binary operation which is
. closed
. has identify
. inverse for every element
* types
 abelian
 finite
 Lie
 nilpotent
* concepts
 representation (matrices)
 commutators
 subgroups
 cosets (left, right)
 simple
 generators
. orbit of a generator
. subgroups generated
. order
. various puzzles
 Cayley graph (?)
 "sameness":
. isomorphic
. homomorpic
 solvable (?)
 products of groups
. Rubik's decomposition
 conjugacy classes
. similar elemets
 connections to solving polynomials
 connections to number theory
* examples (not mutually exclusive; lots more to this list)
 permutation groups
 Top Spin
 other puzzles
 matrix groups
 geometric symmetries
 wallpaper groups
 Escher pictures
 tiling problems
 cube group
 the Monster, "moonshine"
* important theorems
 Sylow
 Lagrange
 classification of finite simple groups
* infinite groups
 Lie groups
 rotations
 boosts
 Lorentz group
 spin, particle representations in physics
Rubik's Cube (also used in variety of stuff above)
* as a group
 generators
 group operation
* possible positions
* God's algorithm
* counting positions
* commutators
* various methods
Physics & Astronomy at Marlboro 
Jim's Schedule
