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 do-hickey 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...
| 1-to-1 |
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".)
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Abelian |
A group G is called abelian if a*b=b*a for all a,b in G.
(See "boring.")
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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).
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Bijection |
A bijection is a map which is both 1-to-1 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} .
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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".)
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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.
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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.
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Cayley's Theorem
(All groups are permutations.) |
Every group G is is isomorphic to a subgroup of SG, 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 Sn.
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Countable |
A set X is countable iff there exists a 1-to-1 map from the positive
integers to X, {1,2,3,...} <-> {X}.
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Commute |
Two group elements a,b are said to commute if a*b=b*a.
(See "Abelian.")
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Commutator |
The commutator [a,b] of two elements is a*b*a-1*b-1.
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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 abelian-like G is.
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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.
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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."
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Cyclic |
The cyclic groups ( Cn ) are those isomorphic to the integers
{0,1,2,3,...,(n-1)} under addition mod n.
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Determinant |
The determinant of an NxN square matrix is the scalar value of
the N-dimensional "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*d-b*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:
- Cn : cyclic group of n elements; order = n
- Sn : permuntations of n objects; order = n!
- An : even permunations of n objects; order = (n!)/2
- Dn : rotations and flips of plane n-gon; 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! 212 8! 38 / 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
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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.
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Function |
A function F(x)=y is a map between two sets, {x}->{y}. (See Map.)
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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).
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| 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.
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Isomorphic |
Two groups G and H are isomorphic if and only if there exists a 1-to-1 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 1-to-1 map f, then f(g1*g2)=f(g1)*f(g2).
Intuitively, isomorphic groups are essentially the same for all practical
purposes.
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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.
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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.)
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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, 1-to-1). 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.)
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Matrices |
There are several ways to define these depending on how picky you
want to get. Technically, matrices are linear maps in an N-dimensional
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 multi-syllabic 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 calculus-level or even
pre-calculus 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.
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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 "1-to-1", "bijection".)
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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, x2, x3, ... }.
For any element x, the orbit of x is a subgroup
of G isorphic to CN, the cyclic group of N elements,
where xN=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 xN=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".
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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.
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Rubik's Cube |
A group-theory permutation puzzle made up of a 3-dimensional array
of smaller "cubies" ( N3 of them, where N=2,3,4,...)
with colored faces. See the java applet at the top of this web page.
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Scalar |
A single real or complex numeric value. (See "Vector", "Matrix".)
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Semi-direct 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
"semi-direct" product of N and M, G = N x| M.
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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 non-abelian 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.)
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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}).
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Subgroup |
A subset of a group which is also a group.
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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")
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