节选Inside Rubik's Cube and Beyond中的一点内容

Cielo

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 C_n (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!3^c2^­e)/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 I_pg, which is obtained by applying g to the start position I_p.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 ]

黑白子

Cielo 发表于 2008-7-3 20:24
Theorem 3 (“second law of cubology”). An operation is possible, if and only if the following three ...

魔方学第二定律是什么？魔方学第一定律又是什么内容？这里的帖子乱码了。另外，什么时间把全书翻译成中文？我很是期待。

黑白子

Cielo 发表于 2008-7-3 19:48
我这里暂时只翻译了前面一点内容：
2.6 特殊子群
三阶魔方群G的子群的结构是多种多样的。

73种不同的阶都是哪些？

黑白子

Cielo 发表于 2008-7-3 20:24
Theorem 3 (“second law of cubology”). An operation is possible, if and only if the following three ...

由于某种原因，有点乱了，能否在整理一下。另外，原书的名字叫什么？有中文版的吗？

Cielo

<P>

原帖由 <I>hqjer</I> 于 2008-7-5 18:58 发表 <A href="http://bbs.mf8-china.com/redirect.php?goto=findpost&amp;pid=176535&amp;ptid=10779" target=_blank><IMG alt="" src="http://bbs.mf8-china.com/images/common/back.gif" border=0></A> 好高深啊 要怎么学习魔方理论呢？

</P>
<P>&nbsp;</P>
<P>我觉得你可以先看看理论区早些时候的帖子，就从理论区最后几页开始看吧<IMG alt="" src="http://bbs.mf8-china.com/images/smilies/default/lol.gif" border=0 smilieid="12"> </P>
<P>&nbsp;</P>
<P>推荐其中pengw、邱志红、rongduo等人的帖子，当然其他作者的帖子也值得一看。</P>

hqjer

好高深啊 要怎么学习魔方理论呢？

kexin_xiao

期待LZ更多的翻译,大家一起研究

bbshanwei

还是翻译过来的好，英文原文的看不懂啊。

pengw

<P>吧里又不止一个魔方理论，建议有异议的人，去细读GGGLGQ关于最小步的循环变换理论，读完全后再细述自已的感受，再找一个魔方去验证其中每一句高论，我可以预言：你读懂了什么也做不了，你没读懂，大师会说你的水平太次，总之一切问题都是你的，哈哈哈。。。</P>
<P>&nbsp;</P>
<P>我为什么要说这么多？看着别人跳坑，有时也很有趣嘛，真是多事。</P>

[ 本帖最后由 pengw 于 2008-7-4 08:30 编辑 ]