The Rubik's Cube group is a
group that represents the
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
of the
Rubik's Cube
The Rubik's Cube is a Three-dimensional space, 3-D combination puzzle originally invented in 1974 by Hungarians, Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik t ...
mechanical puzzle. Each element of the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but also any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position. Indeed with the solved position as a starting point, there is a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between each of the legal positions of the Rubik's Cube and the elements of
.
The group
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
is the
composition of cube moves, corresponding to the result of performing one cube move after another.
The Rubik's Cube group is constructed by labeling each of the 48 non-center facets with the integers 1 to 48. Each configuration of the cube can be represented as a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
of the labels 1 to 48, depending on the position of each facet. Using this representation, the solved cube is the identity permutation which leaves the cube unchanged, while the twelve cube moves that rotate a layer of the cube 90 degrees are represented by their respective permutations. The Rubik's Cube group is the
subgroup of the
symmetric group generated by the six permutations corresponding to the six clockwise cube moves. With this construction, any configuration of the cube reachable through a sequence of cube moves is within the group. Its operation
refers to the
composition of two permutations; within the cube, this refers to combining two sequences of cube moves together, doing one after the other. The Rubik's Cube group is
non-abelian as composition of cube moves is not
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
; doing two sequences of cube moves in a different order can result in a different configuration.
Cube moves
A
Rubik's Cube consists of
''faces'', each with
colored squares called ''facets,'' for a total of
facets. A solved cube has all of the facets on each face having the same color.
A cube move rotates one of the
faces:
or
(half-turn metric).
A center facet rotates about its axis but otherwise stays in the same position.
Cube moves are described with the
Singmaster notation:
The empty move is
. The concatenation
is the same as
, and
is the same as
.
Group structure
The following uses the notation described in
How to solve the Rubik's Cube
How may refer to:
* How (greeting), a word used in some misrepresentations of Native American/First Nations speech
* How, an interrogative word in English grammar
Art and entertainment Literature
* ''How'' (book), a 2007 book by Dov Seidm ...
. The orientation of the six centre facets is fixed.
We can identify each of the six face rotations as elements in the
symmetric group on the set of non-center facets. More concretely, we can label the non-center facets by the numbers 1 through 48, and then identify the six face rotations as elements of the
symmetric group ''S''
48 according to how each move permutes the various facets. The Rubik's Cube group, ''G'', is then defined to be the
subgroup of ''S''
48 generated by the 6 face rotations,
.
The
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of ''G'' is given by
:
.
Despite being this large,
God's Number for Rubik's Cube is 20; that is, any position can be solved in 20 or fewer moves
(where a half-twist is counted as a single move; if a half-twist is counted as two quarter-twists, then God's number is 26).
The largest
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of an element in ''G'' is 1260. For example, one such element of order 1260 is
:
.
''G'' is
non-abelian since, for example,
is not the same as
. That is, not all cube moves
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with each other.
Subgroups
We consider two subgroups of ''G'': First the subgroup ''C''
''o'' of ''cube orientations'', the moves that leave the position of every block fixed, but can change the orientations of blocks. This group is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of ''G''. It can be represented as the normal closure of some moves that flip a few edges or twist a few corners. For example, it is the
normal closure of the following two moves:
:
(twist two corners)
:
(flip two edges).
Second, we take the subgroup
of ''cube permutations'', the moves which can change the positions of the blocks, but leave the orientation fixed. For this subgroup there are several choices, depending on the precise way you define orientation. One choice is the following group, given by generators (the last generator is a 3 cycle on the edges):
:
Since ''C''
''o'' is a normal subgroup and the intersection of ''C''
''o'' and ''C''
''p'' is the identity and their product is the whole cube group, it follows that the cube group ''G'' is the
semi-direct product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in w ...
of these two groups. That is
:
Next we can take a closer look at these two groups. The structure of ''C''
''o'' is
:
since the group of rotations of each corner (resp. edge) cube is
(resp.
), and in each case all but one may be rotated freely, but these rotations determine the orientation of the last one. Noticing that there are 8 corners and 12 edges, and that all the rotation groups are abelian, gives the above structure.
Cube permutations, ''C
p'', is a little more complicated. It has the following two disjoint normal subgroups: the group of even permutations on the corners ''A''
8 and the group of even permutations on the edges ''A''
12. Complementary to these two subgroups is a permutation that swaps two corners and swaps two edges. It turns out that these generate all possible permutations, which means
:
Putting all the pieces together we get that the cube group is isomorphic to
:
This group can also be described as the
subdirect product In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however ne ...
:
,
in the notation of
Griess.
Generalizations
When the centre facet symmetries are taken into account, the symmetry group is a
subgroup of
:
''(This unimportance of centre facet rotations is an implicit example of a
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
at work, shielding the reader from the full
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the object in question.)''
The symmetry group of the Rubik's Cube obtained by disassembling and reassembling it is slightly larger: namely it is the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
:
The first factor is accounted for solely by rotations of the centre pieces, the second solely by symmetries of the corners, and the third solely by symmetries of the edges. The latter two factors are examples of
generalized symmetric groups, which are themselves examples of
wreath products.
The
simple groups that occur as quotients in the
composition series of the standard cube group (i.e. ignoring centre piece rotations) are
,
,
(7 times), and
(12 times).
Conjugacy classes
It has been reported that the Rubik's Cube Group has 81,120
conjugacy classes.
The number was calculated by counting the number of even and odd conjugacy classes in the edge and corner groups separately and then multiplying them, ensuring that the total parity is always even. Special care must be taken to count so-called ''parity-sensitive'' conjugacy classes, whose elements always differ when conjugated with any even element versus any odd element.
See also
*
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
*
Conjugacy class
*
Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
*
Optimal solutions for Rubik's Cube
*
Solvable group
*
Thistlethwaite's algorithm
Notes
References
{{Rubik's Cube
Finite groups
Permutation groups
Rubik's Cube