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Group Theory & Rubik's Cube

Jim Mahoney (


General Info

M,Th 1:30
SciBldg 217
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 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...



(thursdays, for 2 cr folks and 3 cr folks)
  1. for Thurs Jan 20
    1. read "What is GROUP THEORY"
  2. for Thurs Jan 27
    1. read "Examples"
    2. Find the multiplication table for D3, the symmetries (including) "flipping over" of a triangle.
    3. Must it be true that if for elements of a group, if b=a-1, then a=b-1? Give a counter-example, or argue why this must be true. Which group properties does this depend on?
    4. Are the integers a group? Explain.
    5. 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.
  3. for Thurs Feb 3
    1. Find the elements and multiplication table for the symmetry group of a solid 3D cube.
    2. Are any of the groups D4, S4, A4, C4, S5, A5, D5, C5 isometric?
  4. for Thurs Feb 10
    1. Find the commutator subgroup of the solid cube group.
    2. Do R, F generate the cube group? R,F,U? R???
    3. What is the cycle length of the Rubik's commutator FRF'R' ? FR'F'R ?
  5. for Thurs Feb 17
    1. 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?
    2. 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.
    3. Using A and B from the last two problems, find A*B, B*A, [A,B].
    4. Find an element "a" that is conjugate to A and an element "b" which are conjugate to B. (Both elements of S5.)
    5. How many distinct Rubik positions are there 2 moves (quarter turn) from solved? 3 moves? 4?
    6. 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 3-cycles? How are you thinking about what you're doing?
  6. for Thurs Feb 24
    1. 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?
    2. Read Dog School notes, chap 12-14, and be ready to discuss.
  7. for Thurs Mar 2
    1. 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?
    2. How many positions are there in the subgroup generated by only the R and F moves? (the "2-face" subgroup)
    3. See if you can come up with a step-by-step 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 ...
    4. These kinds of solution techniques are called "nested subgroups" method of solution. Why?
  8. Thurs Mar 9
    1. How many distinct groups are there of order 8? Name and/or describe them. (Hint: Lagrange's Theorem provides some assistance.)
    2. Finish the RF algorithms if you haven't already.
    3. Submit an initial proposal for a final project.
    4. Due after break: describe a "solution" to the 3x3 Rubik's Cube.
  9. Thurs Mar 31
    1. Describe a method for "solving" the 3x3 Rubik's Cube.
    2. Stuff you don't have to hand in:
      - Read about "Tilings and Tesselations"; see Resources list below - Play with Kali
  10. Thurs Apr 6
    1. 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.
    2. Pick one of the "star patterns" in the Wallpaper Gallery, and explain which of the 17 wallpaper groups it belongs to and why.
  11. Thurs Apr 13
    1. Peruse the change ringing references
    2. 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...)
  12. Thurs Apr 20
    1. In class: 3D space groups and crystals. References below.
    2. Work on your projects.

Extra Assignments

(occasional; for monday 3 cr folks)
  1. for Mon Feb 7 (originally for 1/31; postponed a week)
    1. Check out first few chapters in yellow Armstrong's Groups and Symmetry.
    2. 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 | cn=1} forms a group. (What group that you already know are these isomorphic to?)
  2. for Mon Feb 14
    1. Read the Housekeeping chapter in the Dog School notes
    2. Read chapters 0 and 1 in Joyner's notes.
    3. Use Venn diagrams to verify the DeMorgan laws (Joyner, chap 0)
    4. Show that group isomorphism is an equivalence relation (Joyner, chap 1)
    5. Show that any equivalence relation implies a partition into disjoint sets.
    6. Show that "conjugacy" (see buzzword definitions) is an equivalence relation.
  3. for Mon Feb 21
    1. Prove that the "solid cube group" is isomorphic to S4. (See Armstrong, chap 8)
    2. Practice some matrix operations if you need to: multiplication, determinant, inverse, as described in class and/or in any standard calculus-level text.
    3. Armstrong, 9.12: Proove that these four matrices form a subgroup of SO3, and find the corresponding rotations.


    4. Armstrong 9.8: Show that these two matrices represent rotations, and find the angle and axis for each:
  4. for Mon Feb 28
    1. 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.)
    2. Show that Cn,m-1 + Cn,m = Cn+1,m, i.e. that Pascal's Triangle works. As usual, Cn,m = n!/(m! (n-m)!) .
    3. Find a 3x3 matrix corresponding to one of the 120 degree corner rotations of the solid cube. Hint: use the 90 face-centered 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 face-centered moves. Also, don't forget the right-to-left vs left-to-right multiplication issue.
  5. 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?)
  6. 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.
  7. Mon Mar 27
    • In class: factor decomposisition of cube group? Are there any nonAbelian simple groups?
  8. 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"
  9. Mon Apr 10
    • Let the matrix that represents rotations by t around the Z axis be
                  [  cos(t)  sin(t)  0 ]
                  [ -sin(t)  cos(t)  0 ]
                  [      0       0   1 ]
      Then as in infinitesimal rotation of an tiny angle dt, this can be written as
                  [ 1 0 0 ]       [  0 1 0 ]
                  [ 0 1 0 ] +  dt [ -1 0 0 ] = 1 + dt Tz
                  [ 0 0 1 ]       [  0 0 0 ] 
      where Tz is the interesting 3x3 matrix shown.
      (a) Make sure you can show all of that, and see how it works.
      (b) What do the analogous matrices Tx and Ty look like?
      (c) Find the "commutator" Tx Ty - Ty Tx.
      (i.e. multiply them in the two different orders and find the difference.)
      We'll talk more about all this stuff next week.
  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.)

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".)
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 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} .

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 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.

Countable A set X is countable iff there exists a 1-to-1 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 abelian-like 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,...,(n-1)} under addition mod n.
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
Equivalence relation An equivalence relation on a set S is a set of pairs of elements (s1,s2) with the following properties:
  1. a~a for all a in S,
  2. a~b imples b~a,
  3. 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 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.

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, 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.)
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:
1 0 0
0 1 0
0 0 1

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.

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

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")



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.
 * 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 

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