An Introduction to Group Theory and the Rubik’s Cube
Lucas Garron
June 8, 2011
1
Introduction and Motivation
If you’ve done enough math or speedcubing, you’ve probably heard that the Rubik’s Cube is a
very useful example for studying the mathematical subject of group theory. The states of the
Rubik’s Cube form a group, which means that they satisfy certain properties; the Rubik’s Cube is
also moderately complicated, which means that a lot of parts of it have interesting properties that
can be used to visualize important concepts in group theory. In particular, the Rubik’s cube group
also contains many subgroups that we can use to understand parts of the cube. This paper is
intended to be a layman’s informal introduction to the Rubik’s Cube Group through a look at its
subgroups. I am assuming that you have some experience with Rubik’s Cubes and mathematical
thinking, but I will try to assume little about what you actually know.
2
2.1
Groups and the Rubik’s Cube
Groups
Groups are not some esoteric and complicated mathematical concept; in fact, the reason gruop
theory is so important is that many mathematical structures you might be familiar with are actually
groups.
1
2.1 Groups
2 GROUPS AND THE RUBIK’S CUBE
Example 1 “The integers under addition” is a description of a common group. Since the set1
{..., −3, −2, −1, 0, 1, 2, 3, ...} is sometimes called Z, this group is often written as (Z, +).
The fact that (Z, +) is a group means that it is possible to add integers to each other, and that
addition in this sense is “well-behaved”. Let’s look at some examples:
• If we add two integers, we always get another integer – the integers are closed under addition.
(If this seems to obvious, notice that 1/2 is not an integer, so we would say the integers are
not closed under division – (Z, ÷) is not a group.)
• If we’re adding 1 + 2 + 3, it also doesn’t make a difference whether we consider it (1 + 2) + 3
or 1 + (2 + 3) – this is because the integers are associative.
• In fact, it doesn’t matter in what order we add two integers: x + y = y + x, no matter what
integers x and y we pick – the integers also happen to be commutative.
• There is also a very important number in the integers: 0. If we add 0 to anything, we always
get back the original number. 0 is called the identity.
• Every integer x also has an inverse −x, and if we add them together we always get the
identity: 0.
Each of these properties might seem obvious, but addition would be more confusing if these
properties were not true – and in fact there are similar mathematical structures that don’t have
some of these properties. However, a group is a structure (a set with an operation) that has the
following properties from above: being closed, associative, having an identity, and having
inverses. Groups also may or may not be commutative.
Definition 1 A group (G, ?) is a set G with an operation ?, such that:
1. Identity: There is an element e such that e ? g = g ? e for every e in G.
2. Closure: If g and h are both in G, then g ? h is also in G
3. Inverses: If g is in G, then there is some other element h such that g ? h = h ? g = e. This
value of h is given the notation g −1 (or −g, when ? is addition).
4. Associativity: If we combine three elements g, h, and k (each in G, then (g ? h) ? k is always
the same as g ? (h ? k)
If g ? h = h ? g is true for any g and h in G, then the group is called commutative or abelian.
1
In mathematics, a set of objects is simply a collection of objects, often written like {2, 5, 20} or
{house, carrot, imagination}. It can be useful to think of it as a “list of things”, except that the order doesn’t
matter – something is either in a certain set or not. For example, {1, 2, 3} and {3, 1, 2, 1} are descriptions of the
same set, because “Is X in the set?” has the same answer for both sets, no matter what X is. Sets can also be
infinite and complicated, so that it wouldn’t make sense to write them down – which is why it’s good to think of a
set in terms of what’s in them and what’s not. For example, “the set of prime numbers” is infinite, but it’s easy to
tell whether something is in the set or not, assuming we can tell what a prime number is.
2
2.1 Groups
2 GROUPS AND THE RUBIK’S CUBE
For the integers, we can see that G = Z, “ ? ” = “+00 , and e = 0, making these statements agree
with our earlier look at the integers.
There is also one more important concept which shows up commonly in the study of the cube:
the order of a group or an element.
Definition 2 The order of a group (G, ?) is the number of elements in G, written as |G|.
Definition 3 The order of an element g in a group (G, ?) is the smallest number of times we
have to repeat it to get back to the identity. If we write g ? g ? ... ? g as g n , then the order of g is
{z
}
|
n
the smallest positive integer i such that g i = e. We write the order of g as |g|.
(In case you’re wondering, the reason that these definitions use the same word is that the
powers g 0 , g 1 , ... form a subgroup of of G generated by g which has the same order as g. This
should become more clear when we define those terms below.)
It might be the case that an element doesn’t have an order; for example, 1 doesn’t have an
order in (Z, +); no multiple of 1 is equal to 0. In this case, we often say that the order of 1 is
infinite. However, the following is true:
Theorem 1 If (G, ?) is a finite group (i.e. G has only finitely many elements), then every element
has an order.
Proof: Since G is finite, it can’t be the case that g 0 , g 1 , g 2 , g 3 , ... are all different, so there
must be some element g i that is equal to another element g j that comes after it. Using some
implicit, but hopefully logical, notation: g −i is the inverse of g i , so we can “unwind” to show that
g j−i = g j g −i = g i g −i = e. Thus, j − i (which is positive) is an upper bound for the order of g. It
might be the order, or the order is some smaller number.
Since the Rubik’s Cube has a finite number of states, this means that every algorithm has an
order. For example |R| = 4. However, we’re getting ahead of ourselves; to understand what this
all should mean, let’s take a look at the Rubik’s Cube.
(a) A Rubik’s Cube in
its unique solved state.
(b) A solved Rubik’s
Cube after one move
performed on its right
side: R.
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(c) An arbitrary arrangement of the cube,
reached by performing
successive turns.
2.2 A Crash Course in Rubik’s Cube Terminology and2 Notation
GROUPS AND THE RUBIK’S CUBE
2.2
A Crash Course in Rubik’s Cube Terminology and Notation
A 3x3x3 Rubik’s Cube has 26 cubies with 54 stickers on 6 faces (9 stickers), which we call {back,
front, up, down, left, right}, abbreviated respectively as {B, F, U, D, L, R}. Of these cubies, there
are 8 edges, 12 corners, and 6 centers. Each side is composed of 9 “cubies” (4 edges, 4 corners,
and 1 center).
A Rubik’s Cube has 432520032744898560002 distinct states that can be reached by performing
moves on the sides of the cube.
If we look the side R and turn it clockwise around the center, then this is also the move denoted
R. If we turn it counter-clockwise, the move is R0 , and if we turn it either way twice it is R2.
Exactly the same holds for other sides in place of R. Note that “do an R move followed by a U 0
move” is traditionally denoted as R U 0 when we are studying the cube. Such a move sequence is
also called an algorithm. These definitions are contrary to mathematical convention, but they
are clearly established for the study of the Rubik’s Cube.
2.3
The Rubik’s Cube
I stated earlier that the Rubik’s Cube is a group. The states of a Rubik’s cube are a set, which
we’ll call G (and which has order |G| = 43252003274489856000). but if we want an operation ?,
what should it in this case? How do you add two states? One reasonable suggestion might be to
consider what happens if we perform some moves on a cube. Lets say we perform some algorithm
A that puts the cube into some state X, and another algorithm B that puts the cube into state Y .
It would be nice if “X ? Y ” happened to be the state you get if you perform the algorithms after
each other, A B. Since every position can be created by some sequence of moves, this allows us
to assume that you can add any two positions, even if you don’t know what moves might create
those positions. It turns out that this exactly defines the group (G, ?).
As with any such definition, we should check whether it is well-defined. There are multiple
algorithms for each state, and it is not guaranteed that the definition always gives us the same
“sum”. To see why this is an issue, consider the algorithms T1 = RU R0 U 0 R0 F R2U 0 R0 U 0 RU R0 F 0
and T2 = R2U 0 R2U R2U D0 R2U R2U 0 R2D on the 4x4x4 cube. Both of these seem to be algorithms
for the same state – the so-called T-Permutation. Now consider the algorithm r (an inner layer),
and what happens if we perform r followed by a T-Permutation: it matters which T-Permutation
we use, because the center pieces are affected differently! r ? T1 6= r ? T2 ; the action is not faithful
unless we can distinguish all the cubies, e.g. when the center cubies are labeled on a supercube.
Therefore, the states of the 4x4x4 cube do not form a group unless we are careful about the
definition of “state”.
It turns out that this is not an issue on the 3x3x3 cube. This cube is small enough (each
cubie) that it is possible to tell where every sticker in a state came from. Any algorithms for
the same state move the stickers in the same permutation. Therefore, on a 3x3x3 cube it is
justifiable to think of RU R0 U 0 R0 F R2U 0 R0 U 0 RU R0 F 0 and R2U 0 R2U R2U D0 R2U R2U 0 R2D, as well
as the resulting state, all as representations of the T-Permutation. A state and an algorithm
describe exactly the same thing, although there are only a finite number of states, and an infinite
number of algorithms for each state. Both algorithms result in the same state starting from solved,
2
It is an amusing curiosity that the number of unsolved states of the Rubik’s Cube, 43252003274489855999,
is a prime number, as observed by Chris Hardwick and discussed in June of 2007. By the prime number theorem,
this is not particularly likely. I found this property to hold for the 3x3x3, 8x8x8, and 11x11x11 cube, but no others
until 40x40x40.
4
2.4 Getting Specificational: Permutations
2 GROUPS AND THE RUBIK’S CUBE
and it is guaranteed that “applying” these algorithms to other states will always give the same
result. In fact, we can omit the ? symbol because composing moves by writing them one after the
other is well-defined. In group theory, the operation of such a group (where we take an element
and apply it to another one to get a new element) is often called a composition. This is why
cube theorists sometimes use descriptions like “the superflip position composed with four dots”.
It is possible to think about cube moves mainly as permutations, but it is often nice to understand
that we could always just “do the moves” because the cube is a group. For example, R is a 1-move
algorithm, and R R0 is an alternate description for the solved state
Now, our definitions for the Rubik’s Cube satisfy all the properties of a group:
• The set of elements, G, is simply the set of Rubik’s Cube states (all of which which can be
described by multiple algorithms), and ? is composition
• Identity: e is the solved state. Composing the solved state with something is idempotent (it
does nothing to the other state).
• Closure: Two states can always be composed.
• Inverses: Every state has an inverse. For example, the inverse of (RU R0 F 0 U 2) is (U 2F RU 0 R0 ).
However, (L0 U 0 LF U 2) is also an inverse. Thus it must be the case that (RU R0 F 0 U 2) =
(L0 U 0 LF U 2); both of these are descriptions of the inverse state.
• Associativity: If we have compose three states as A1 A2 A3 , it makes no difference if we
compute an algorithm for A2 A3 and apply it to A1 , or if we apply A3 to an algorithm for A1 ;
the result is the same.
2.4
Getting Specificational: Permutations
Although it is good to establish the intuitive grounds behind the definitions, it is important to
understand how puzzles like the Rubik’s Cube are mathematically defined, and that is through
permutations. A permutation is an operation that takes some arrangement elements, and shuffles
them around. For example, when we scramble a Rubik’s Cube we move its 26 cubies around, and
when we are done each cubie has moved to some new position (possibly the same). We say that
our scramble induces a permutation on the cubies. Similarly, it induces a permutation on the
stickers of the cube. Ignoring the centers of each side (because they never move), the Rubik’s Cube
(a) The result of either T-Permutation algorithm
(b) The algorithm r
(c) The algorithm (r) ?
(R U R’ U’ R’ F R2 U’
R’ U’ R U R’ F’)
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(d) The algorithm (r) ?
(R2 U’ R2 U R2 U D’
R2 U R2 U’ R2 D)
2.4 Getting Specificational: Permutations
2 GROUPS AND THE RUBIK’S CUBE
has 48 stickers that can move. If we perform the move R, then the sticker at position 25 moves
to the one at position 27, which moves to 32, which moves to 30, which moves back to 25. This
is called a cycle of stickers, and it would be written as (25 27 32 30). It should make sense that
every state can be described in terms of how it moves the stickers. For example, R induces the
permutation with the following cycle structure:
R = (25 27 32 30)(26 29 31 28)(3 38 43 19)(5 36 45 21)(8 33 48 24)
We can figure out where any sticker goes by looking at the cycles: sticker 38 moves to the
position of sticker 43; sticker 28 to 26 (since a cycle permutes the stickers in a loop), and sticker 20
stays in place because it is not moved by any cycle. Instead of writing down the cycle structure,
it would also have been useful
to write down the actual permutation map (1 7→ 1, 2 7→ 2, 3 7→
1 2 3 4 5 ...
38, 4 7→ 5, 7→ 36, ...), or
that tells us where each sticker goes, but this
1 2 38 4 36 ...
takes up more space.
It should now make more sense to say what a state is: A state of the Rubik’s Cube is a
permutation of stickers. An algorithm that turns a solved cube into that state will induce exactly
the same permutation of stickers. If we want to compose two permutations, we simply figure
out where each sticker is moved by the first permutation, then where the new position of that
sticker is moved by the second permutation. Thus, R U is the state defined by the permutation
that results when we compose R with U – which exactly corresponds to how the stickers permute
when we apply those two moves. However, we can now do something cooler: If we have two
arbitrary states of the Rubik’s Cube, we can compose them just by composing their successive
permutations, without having to find an algorithm for either one! That we can also do this
through algorithms (such that the answer is well-defined) comes from the fact that an algorithm is
really just an explicit composition of the permutations that are basic moves of the Rubik’s Cube.
To be precise, the permutations of the stickers are not the Rubik’s Cube group; instead, the
Rubik’s Cube group acts on the set of stickers, and the permutations of this group action give
us full information about the Rubik’s Cube group.
1
2
3
4
U
5
6
7
8
9
10 11 17 18 19 25 26 27 33 34 35
12
L
13 20
F
21 28
R
29 36
B
37
14 15 16 22 23 24 30 31 32 38 39 40
41 42 43
44
D
45
46 47 48
(a) Labeled Stickers in a flattened view
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3 SUBGROUPS
3
Subgroups
A subgroup is simply a group that is contained in another group. We can use the definition of a
group to find out the criteria for a subgroup.
Definition 4 (H, ?) is a subgroup of (G, ?) if H is a subset of G (that is, all the elements of H
are also in G), and the following are true:
• e is in H, since every group must have an identity, and e is the only candidate in G
• H is closed: if h1 and h2 are in H, then h1 ? h2 must also be in H for H to be a group.
• H has inverses: for any h in H, the inverse of h must also be. Else, again, the corresponding
condition of a group is violated.
• The elements of H are associative: This is automatically true, since
Proposition 1 If R is in a certain subgroup of the Rubik’s Cube, so are R2 and R0 .
Proof: Since a subgroup is closed, R ? R = R2 must also be in the group. The inverse, R0 ,
must also be in the group.
What if we consider a subgroup containing both R and U ? It must be the case that R2 and R0
are in the subgroup, as well as R U and U 0 R2... In fact, any algorithm composed of U -turns and
R-turns must be in the subgroup. There may be other states in the subgroup (we didn’t say L is
not in the group), but there are at least some elements that must be guaranteed to be in such a
subgroup. In fact, mathematicians have a name for “the smallest subgroup containing R and U ”;
it is called “the subgroup generated by R and U ”.
Definition 5 Given a set of elements of a group G (e.g. cube states) g1 , g2 , g3 , ..., the smallest
subgroup containing all these elements is denoted hg1 , g2 , g3 , ...i. Such a subgroup always exists.
Proof: Note that such a set is always guaranteed to exist, since G is such a set. However, it
is not immediate that this set is a subgroup. However, let’s consider all subgroups that contain
g1 , g2 , g3 , .... There must be some minimum set that they all contain (their intersection); let’s
call it X. If we go through the conditions of a group again, we can show that X must be a group:
• X contains the identity, since it is an element of all these subgroups.
• X is closed: If g and h are in X, they are in every one of these subgroups, as is g ? h
• X contains the inverses of all its elements. Similarly, if g is in X, it is in all these subgroup,
as is gı – thus gı is in X.
• X is obviously associative, since the original group was.
Thus, by definition, X “forces” itself to be a group, and it is unambiguous. to refer to a
subgroup generated by any arbitrary elements.
Theorem 2 Any subgroup can be written using generators.
Proof: If we take a subgroup, and consider the subgroup generated by its elements, we obviously get. . . itself. There may be a shorter representation, but there is guaranteed to be a
presentation using generators.
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3.1 Examples of Rubik’s Cube Subgroups
3.1
3 SUBGROUPS
Examples of Rubik’s Cube Subgroups
Here are some examples of subgroups of the cube:
• Cubers will be familiar with hR, U i as a two-gen subgroup.
• The so-called PLLs (Permutations of the Last Layer, preserving sticker orientation) form
a subgroup. It can be written as h U , R0 U R0 U 0 R0 U 0 R0 U RU R2 , RB 0 RF 2R0 BRF 2R2 i,
which might be characterized as “the PLLs are generated by U , the U b-Permutation, and
the Ab-Permutation”. (If you have a cube in front of you, try it out! Set up an arbitrary
PLL and check that it can be solved by U b and Ab and AUFs (U -turns).)
• hei is the trivial subgroup. It might seem silly, but it it is good to know by name. For
example, the trivial subgroup is a subgroup of every other subgroup.
• Every valid state of the Rubik’s Cube can be reached by turning the six sides. Thus, G =
hB, F, U, D, L, Ri.
• It is possible to generate any move using the 5 others, as with D = (L0 RF 2B2L0 R)U (L0 RF 2B2L0 R).
Thus, it is also true that G = hB, F, U, L, Ri, since any state has algorithm using all six generators that can be rewritten by replacing D using that algorithm. Informally, we might say
that the cube is five-gen.
• It is a little-known fact (among cubers) that the entire Rubik’s Cube Group is actually twogen. For example, G = hU BLU L0 U 0 B 0 , R2F LD0 R0 i. There are some interesting properties
which were discussed on the speedsolving.com forum in March 2010. In particular, Tom
Rokicki found that if you take any two random states of the cube, there is about a 50%
chance that these two states generate the entire group. It is also possible to generate the
cube using two elements of orders as low as 2 and 4; this demonstrates how easily two
algorithms can “tangle”.
• If we take any state X of the Rubik’s Cube, then hXi is a subgroup. Notice that it is
always commutative, since all the elements can be written as X i for some integer i, and
that |X| = |hXi|. It is sometimes unclear to cubers that repeating an algorithm repeatedly
will always return to the original state, but this is a simple fact from a group-theoretic
perspective.
We can ask what possible subgroups we can get from this, and the most important question
to answer is what possible orders subgroups generated by one element can have (note from
above that two elements can already generate the entire group). By analyzing all possible
cycle decompositions of states, we can show that there are 73 possible orders:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22,
24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 66, 70, 72,
77, 80, 84, 90, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 165, 168, 180, 198,
210, 231, 240, 252, 280, 315, 330, 336, 360, 420, 462, 495, 504, 630, 720, 840, 990, 1260}
The solved state is obviously the only state with order 1. There are many states with order
1260, such as RF 2B 0 U B 0 . There was a discussion in August 2010 that covers a little more
8
3.1 Examples of Rubik’s Cube Subgroups
3 SUBGROUPS
of this. It is also known how many states of each order, as reported in the Cubic Circular by
David Singmaster in 1982, but the combinatorics behind this are more complicated.
• Sylow Subgroups are important subgroups of any group. The order of the Rubik’s Cube
factors as:
|G| = 43252003274489856000 = 227 · 314 · 53 · 72 · 111
One of the Sylow’s theorems states that G contains certain Sylow subgroups of each of the
following orders: 22 7, 31 4, 53 , 72 , 111 , and that the Sylow subgroups of any given order are
all “similar” in an important sense (they are conjugate). I constructed explicit generators
for these groups in March 2010.
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