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2.6 Special Subgroups
The subgroup structure of Rubik’s group G is extremely varied.
The easiest way, for the present, is to find all the cyclicsubgroups of G. A group is called cyclic,if it is generated by one single element. Since every finite cyclic group of the order n is isomorphic to Cn (Example2.2.2) and every infinite cyclic group is isomorphic to the additive group of integers (Example 2.2.1) we already know the structure of all cyclic groups. By the order of an element a of a group A we mean the order of the cyclic subgroup generated by a. In the case of a finite group this is the smallest natural number n with an = e (neutral element). For Rubik’s group the order of all the elements can be immediately read off the cyclic decomposition: It is the least common multiple of the cycle lengths multiplied by 3 (twisting corner cycles),or by 2 (reorienting edge cycles), or by 1 (orientation-preserving cycles).There exist precisely 73 different orders and maximum order is 2·2·3·3·5·7 = 1260. The following short maneuver for an operation of this order has been found by J. B. Butler:
RU2D’BD’ (5) ->(-ufl, lbu, rfu)(+ubr,fdl,dfr,rbd,ldb)
(+uf,lb, dr, fr, ul, ur, bu)(+dl, rb)(df, db)
By the way, 1260 is also the maximum orderin the group G\bar, and here a maneuver with one single layer move is already sufficient:
RCu (1) ->(+ufl, ulb,ubr, rdf)(+urf)(+dlf, dbl, drb)
(+uf,ul, ub, ur, rf)(+df, fl, fb, dr, lf, bl, rb)
(f,l, b, r)
In general, we are particularly interested in subgroups defined by either the requirement not to move certain cubies, or to move them only in a restricted way, or by a restriction to certain moves or maneuvers. It follows from the second law of cubology (Theorem 2.4.3) that the structure of the subgroup of all possible operations which leave a certain subset C of the set of all corner cubies and acertain subset E of the set of alledge cubies untouched (elementwise fixed), does not depend on the location but only on the number of the corner and edge cubies remaining untouched. With c := 8 - |C| and e := 12 - |E|, such a subgroup has the order (c!e!3c2e)/12. As an example of asubgroup defined by a restriction to certain moves, we look at the “squaregroup” <<R2, L2, F2, B2,U2, D2>> generated by the operations of the six squaremoves R2, L2, F2, B2,U2, D2. (The inner brackets are supposed to indicate thetransition from the maneuvers to the operations, i.e. the homomorphism π, while the outer brackets indicate the transition to the generatedsubgroup). As already frequently done, we identify every operation g with the position Ipg, which is obtained by applying g to the start position Ip.We call a cubie red or blue etc., if one of its color tiles is red or blue etc.Colors sitting opposite each other in the start position are called “countercolors”.
[ 本帖最后由 Cielo 于 2009-4-14 20:04 编辑 ] |
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