A Subgroup R 2 L of The Rubik S Cube
A Subgroup R 2 L of The Rubik S Cube
A Subgroup R 2 L of The Rubik S Cube
A group is a nonempty set of elements along This subgroup is Abelian because the order of
Our subgroup is generated by the elements RR and L.
with a binary operation that satisfy the follow- its elements does not change the result.
ing: closure under the group operation, asso- This subgroup is not cyclic because no element
ciativity, an identity element exists, and every G= , of this subgroup will generate the entire sub-
element has an inverse element. If the elements group.
in a group commute, then we say that the group The elements of G are: The orders of the elements are: |I| = 1, |R2|
is abelian. = 2, |L| = 4, |L2| = 2, |L3| = 4, |R2L| = 4,
|R2L2| = 2, and |R2L3| = 4.
Cayley Table Although we have 8 elements in our group, L =
Subgroups L3 and R2L = R2L3, therefore, we only have
6 subgroups.
In general if a nonempty subset H of a group G
is itself a group under the operation of G, we
say that H is a subgroup of G. In this case we Subgroup Diagram
have a subset of moves (R2, L) as being allow-
able learn to solve cubes that were jumbled with
only those moves and by doing that we are effec- ,
tively simply reducing the number of allowable PPP
permutations. PP
PP
PP
PP
PP
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Z
The Rubik’s Cube can be viewed as a group be- Z
Z
Z
cause it is a nonempty set of six distinct col- T
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Z
Z
lection of moves under combinations of moves.
T Z
T Z
T Z