|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Group theory sidebar}}
| | Hello, my title is Andrew and my wife doesn't like it at all. It's not a common thing but what I like performing is to climb but I don't have the time recently. Her family members life in Ohio but her spouse desires them to transfer. He is an purchase clerk and it's something he really appreciate.<br><br>My web site :: [http://chungmuroresidence.com/xe/reservation_branch2/152663 online psychic reading] |
| | |
| The '''Rubik’s Cube group''' is a [[Group (mathematics)|group]] <span class=nowrap>(''G'', •)</span> that corresponds to the [[Set (mathematics)|set]] ''G'' of all ''cube moves'' on the [[Rubik's Cube]] mechanical puzzle with the group [[Binary operation|operation]] • being the concatenation of cube moves. With the solved position as a starting point, there is a [[Bijection|one-to-one correspondence]] between each of the legal positions of the Rubik's Cube and the elements of ''G''.<ref name="advgroup">{{cite book |author=Joyner, David |title=Adventures in group theory: Rubik's Cube, Merlin's machine, and Other Mathematical Toys |publisher=Johns Hopkins University Press |year=2002 |isbn=0-8018-6947-1}}</ref><ref name="geometer">{{cite web |author=Davis, Tom |url=http://geometer.org/rubik/group.pdf |year=2006 |title=Group Theory via Rubik’s Cube}}</ref>
| |
| | |
| ==Cube moves==
| |
| | |
| A 3×3×3 Rubik's Cube consists of 6 ''faces'', each with 9 colored squares called ''facets,'' for a total of 54 facets. A solved cube has all of the facets on each face having the same color.
| |
| | |
| A cube move rotates one of the 6 faces 90°, 180° or -90° (half move metric).<ref name="cube20"/> A center facet rotates about its axis but otherwise stays in the same position.<ref name="advgroup" />
| |
| | |
| Cube moves are described with the [[David Singmaster|Singmaster]] notation:<ref name = "Singmaster">{{cite book|last=Singmaster|first=David|title=Notes on Rubik's Magic Cube|year=1981|publisher=Penguin Books|isbn=0907395007}}</ref>
| |
| {| class="wikitable" style="margin: 1em auto 1em auto;"
| |
| |- style="text-align:center;"
| |
| | '''Basic 90°'''
| |
| | '''180°'''
| |
| | '''-90°'''
| |
| |-
| |
| | <math>F</math> turns the front clockwise
| |
| | <math>F^2</math> turns the front clockwise twice
| |
| | <math>F^\prime </math> turns the front counter-clockwise
| |
| |-
| |
| | <math>B</math> turns the back clockwise
| |
| | <math>B^2</math> turns the back clockwise twice
| |
| | <math>B^\prime </math> turns the back counter-clockwise
| |
| |-
| |
| | <math>U</math> turns the top clockwise
| |
| | <math>U^2</math> turns the top clockwise twice
| |
| | <math>U^\prime </math> turns the top counter-clockwise
| |
| |-
| |
| | <math>D</math> turns the bottom clockwise
| |
| | <math>D^2</math> turns the bottom clockwise twice
| |
| | <math>D^\prime </math> turns the bottom counter-clockwise
| |
| |-
| |
| | <math>L</math> turns the left face clockwise
| |
| | <math>L^2</math> turns the left face clockwise twice
| |
| | <math>L^\prime </math> turns the left face counter-clockwise
| |
| |-
| |
| | <math>R</math> turns the right face clockwise
| |
| | <math>R^2</math> turns the right face clockwise twice
| |
| | <math>R^\prime </math> turns the right face counter-clockwise
| |
| |}
| |
| | |
| The empty move is <math>E</math>. The concatenation of <math>LLLL</math> is the same as <math>E</math>, and <math>RRR</math> is the same as <math>R^\prime</math>.
| |
| | |
| ==Group structure==
| |
| The following uses the notation described in [[wikibooks:How to solve the Rubik's Cube|How to solve the Rubik's Cube]]. 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''<sub>48</sub> according to how each move permutes the various facets. The Rubik's Cube group, ''G'', is then defined to be the [[subgroup]] of ''S''<sub>48</sub> [[group generator|generated]] by the 6 face rotations, <math>
| |
| \{F,B,U,D,L,R\}</math>.
| |
| | |
| The cardinality of ''G'' is given by <math>|G| = 43{,}252{,}003{,}274{,}489{,}856{,}000\,\! = 2^{27} 3^{14} 5^3 7^2 11</math>.<ref name="GAP">{{cite web|last=Schönert|first=Martin| url=http://www.gap-system.org/Doc/Examples/rubik.html|title=Analyzing Rubik's Cube with GAP}}</ref>
| |
| Despite being this large, any position can be solved in 20 or fewer moves (where a half-twist is counted as a single move).<ref name="cube20">{{cite web|url=http://www.cube20.org|author=Rokicki, Tomas et al|title=God's Number is 20}}</ref>
| |
| | |
| The largest [[Cyclic group|order]] of an element in ''G'' is 1260. For example, one such element of order 1260 is <math>(RU^2D^{-1}BD^{-1})</math>.<ref name="advgroup"/>
| |
| | |
| ''G'' is [[Non-abelian group|non-abelian]] since, for example, <math>FR</math> is not the same as <math>RF</math>. That is, not all cube moves [[Commutativity|commute]] with each other.<ref name="geometer" />
| |
| | |
| We consider two subgroups of ''G'': First the group of cube orientations, ''C''<sub>''o''</sub>, which leaves every block fixed, but can change its orientation. This group is a [[normal subgroup]] 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:
| |
| | |
| :<math>B R^\prime D^2 R B^\prime U^2 B R^\prime D^2 R B^\prime U^2,\,\!</math> (twist two corners)
| |
| :<math>R U D B^2 U^2 B^\prime U B U B^2 D^\prime R^\prime U^\prime,\,\!</math> (flip two edges).
| |
| | |
| For the second group we take ''G'' permutations, ''C<sub>p</sub>'', which can move the blocks around, but leave the orientation fixed. For this subgroup there are more choices, depending on the precise way you fix the orientation. One choice is the following group, given by generators (the last generator is a 3 cycle on the edges):
| |
| | |
| :<math>C_p = [U^2, D^2, F, B, L^2, R^2, R^2 U^\prime F B^\prime R^2 F^\prime B U^\prime R^2].\,\!</math>
| |
| | |
| Since ''C''<sub>''o''</sub> is a normal subgroup, the intersection of ''C''<sub>''o''</sub> and ''C''<sub>''p''</sub> is the identity, and their product is the whole cube group, it follows that the cube group ''G'' is the [[semi-direct product]] of these two groups. That is
| |
| | |
| :<math> G = C_o \rtimes C_p. \, </math>
| |
| | |
| ''(For technical reasons, the above analysis is not complete. However, the possible permutations of the cubes, even when ignoring the orientations of the said cubes, is at the same time no bigger than ''C<sub>p</sub>'' and at least as big as ''C<sub>p</sub>'', and this means that the cube group is the semi-direct product given above.)''
| |
| | |
| Next we can take a closer look at these two groups. The structure of ''C''<sub>''o''</sub> is
| |
| | |
| :<math>\mathbb Z_3^7 \times \mathbb Z_2^{11},\ </math>
| |
| | |
| since the group of rotations of each corner (resp. edge) cube is <math>\mathbb Z_3</math> (resp. <math>\mathbb Z_2</math>), 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<sub>p</sub>'', is a little more complicated. It has the following two normal subgroups, the group of even permutations on the corners ''A''<sub>8</sub> and the group of even permutations on the edges ''A''<sub>12</sub>. Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that
| |
| | |
| :<math>C_p = (A_8 \times A_{12})\, \rtimes \mathbb Z_2.</math> | |
| | |
| Putting all the pieces together we get that the cube group is isomorphic to
| |
| | |
| :<math>(\mathbb Z_3^7 \times \mathbb Z_2^{11}) \rtimes \,((A_8 \times A_{12}) \rtimes \mathbb Z_2).</math> | |
| | |
| This group can also be described as the [[subdirect product]] <math>[(\mathbb Z_3^7 \rtimes \mathrm S_8) \times (\mathbb Z_2^{11} \rtimes \mathrm{S}_{12})]^\frac{1}{2}</math>, in the notation of [[Robert Griess|Griess]]{{cn|date=September 2013}}.
| |
| | |
| ===Generalizations===
| |
| When the centre facet symmetries are taken into account, the symmetry group is a [[subgroup]] of
| |
| | |
| : <math>[\mathbb Z_4^6 \times (\mathbb Z_3^7 \rtimes \mathrm S_8) \times (\mathbb Z_2^{11} \rtimes \mathrm S_{12})]^\frac{1}{2}.</math> | |
| | |
| ''(This unimportance of centre facet rotations is an implicit example of a [[quotient group]] at work, shielding the reader from the full [[automorphism group]] of the object in question.)''
| |
| | |
| The symmetry group of the Rubik's Cube obtained by dismembering it and reassembling is slightly larger: namely it is the [[direct product of groups|direct product]]
| |
| | |
| : <math>\mathbb Z_4^6 \times \mathbb Z_3 \wr \mathrm S_8 \times \mathbb Z_2\wr \mathrm S_{12}.</math>
| |
| | |
| 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 [[wreath product]]s.
| |
| | |
| The [[simple group]]s that occur as quotients in the [[composition series]] of the standard cube group (i.e. ignoring centre piece rotations) are <math>A_8</math>, <math>A_{12}</math>, <math>\mathbb Z_3</math> (7 times), and <math>\mathbb Z_2</math> (12 times).
| |
| | |
| ==Citations==
| |
| {{reflist|30em}}
| |
| | |
| ==See also==
| |
| {{Div col|cols=3}}
| |
| *[[Commutator]]
| |
| *[[Conjugacy class]] -Conjugates
| |
| *[[Coset]]
| |
| *[[Optimal solutions for Rubik's Cube]]
| |
| *[[Solvable group]]
| |
| *[[Morwen Thistlethwaite#Thistlethwaite's algorithm| Thistlethwaite's algorithm]]
| |
| {{Div col end}}
| |
| {{Rubik's Cube}}
| |
| [[Category:Finite groups]]
| |
| [[Category:Permutation groups]]
| |
| [[Category:Rubik's Cube]]
| |
Hello, my title is Andrew and my wife doesn't like it at all. It's not a common thing but what I like performing is to climb but I don't have the time recently. Her family members life in Ohio but her spouse desires them to transfer. He is an purchase clerk and it's something he really appreciate.
My web site :: online psychic reading