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 ]

Cielo

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。
现在看来，研究G的子群的最简单方法，就是去寻找所有的G的循环子群。一个群，如果是由单独一个元素生成的，就称它为循环的。任意一个n阶有限循环群都同构于C_n（见例2.2.2），任意一个无限循环群都同构于整数的加法群（例2.2.1），这样我们已经知道了所有循环群的结构。而群A中的一个元素a的阶，定义为这个元素a生成的子群的阶。在有限群的情况下，a的阶n是满足下面这个式子的最小的自然数：aⁿ = e（这里e是群的单位元）。对于三阶魔方群，其任意一个元素的阶可以方便地通过分解为若干循环而看出来：它就是下面这些数的最小公倍数，即循环的长度乘以3（如果该循环整体效果包含了角块的扭转）、或者乘以2（如果该循环整体效果包含了棱块的翻转）、或者乘以1（如果该循环保持色向）。一共有73种不同的阶，而其中最大的阶是2•2•3•3•5•7 = 1260。下面这个1260阶的操作是J. B. Butler发现的：
RU2D’BD’ (5) ->       (-ufl, lbu, rfu)(+ubr,fdl,dfr,rbd,ldb)
                                (+uf, lb, dr, fr, ul, ur, bu)(+dl, rb)(df, db)
顺便说一句，1260也是群G\bar（原书中是G上方加一个横线，但我不会用word打出那个符号）的元素的最大阶，而且这里有个策略只动了一层就足够使阶为1260了：
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)
总体说来，我们特别感兴趣的是有下面若干限制的子群，比如规定了不能动哪些块或者只能动哪些块。


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感谢楼下乌木先生指出了我打错的地方，已改正！</P>

[ 本帖最后由 Cielo 于 2009-4-14 20:11 编辑 ]

Cielo

<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><FONT color=#000000><FONT face="宋体, MS Song"><B style="mso-bidi-font-weight: normal"><SPAN lang=EN-US>Theorem 3 (“second law of cubology”). </SPAN></B><SPAN lang=EN-US>An operation is possible, if and only if the following three conditions are fulfilled:</SPAN></FONT></FONT></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><SPAN lang=EN-US><FONT face="宋体, MS Song" color=#000000>(a) The total number of cycles of even length (corner and edge cycles) is even.</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><SPAN lang=EN-US><FONT face="宋体, MS Song" color=#000000>(b) The number of right-twisting corner cycles is equal to the number of letf-twisting corner cycles modulo 3.</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><SPAN lang=EN-US><FONT face="宋体, MS Song" color=#000000>(c) The number of reorienting edge cycles is even.</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><FONT color=#000000><B style="mso-bidi-font-weight: normal"><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri">定理</SPAN><SPAN lang=EN-US><FONT face="宋体, MS Song">3</FONT></SPAN></B><B style="mso-bidi-font-weight: normal"><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri">（“魔方学第二定律”）</SPAN></B><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri">一个操作<SUP>①</SUP>是可能的，当且仅当下面三条同时满足：</SPAN></FONT></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri"><FONT color=#000000>一、所有的长度为偶数的环（包括偶数角块环和偶数棱块环）的个数是偶数，</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri"><FONT color=#000000>二、效果为顺时针扭转的角块环的个数与效果为逆时针扭转的角块环的个数相等，</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt; mso-pagination: widow-orphan"><SPAN style="FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri"><FONT color=#000000>三、被翻转了的棱块的个数是偶数。</FONT></SPAN></P>
<P><FONT color=#000000><SPAN style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri; mso-bidi-font-size: 11.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">注<SUP>①</SUP>：这里的“操作”不是我们常说的某个公式，而是指某个公式所引起的块的位置和方向的变化。原书中代表“公式”是这个词“</SPAN><SPAN lang=EN-US style="FONT-SIZE: 10.5pt; FONT-FAMILY: Calibri; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: 宋体; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"><FONT face="宋体, MS Song">maneuver</FONT></SPAN><SPAN style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-ascii-font-family: Calibri; mso-hansi-font-family: Calibri; mso-bidi-font-size: 11.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">”。</SPAN></FONT></P>
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<P>忍大师的N阶定律说的也是同一个意思。</P>

Cielo

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原帖由 <I>kexin_xiao</I> 于 2008-7-3 20:22 发表 <A href="http://bbs.mf8-china.com/redirect.php?goto=findpost&amp;pid=175044&amp;ptid=10779" target=_blank><IMG alt="" src="http://bbs.mf8-china.com/images/common/back.gif" border=0></A> 太专业了<IMG alt= src="http://bbs.mf8-china.com/images/smilies/default/handshake.gif" border=0 smilieid="17"> <IMG alt= src="http://bbs.mf8-china.com/images/smilies/default/sweat.gif" border=0 smilieid="10">

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<>有时间的话我会再多翻译一点内容，尽管翻译得不好，肯定有很多不准确的地方……</P>
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<>我觉得这本书里写这些数学中群论中的术语其实没必要，我最近在看这本书，但到现在也没看多少，看里面定理的证明也没有用到很多数学的东西，毕竟是二十多年前的书了<IMG alt="" src="http://bbs.mf8-china.com/images/smilies/default/sweat.gif" border=0 smilieid="10"> </P>
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Cielo

<P>呵呵，这本来就是二十几年前的书，当然没有我们所期望的新东西啦。或者甚至可以这样说，我们现在所知道的与这二十几年前的相比，只怕也没多出什么新东西吧。</P>
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<P>此外当然也不能保证这书上写的就都对，所以不存在外国的月亮圆的问题。</P>
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<P>我个人的感觉也是根本不用扯上那么多群论的术语，基本原理只是一些排列组合而已。</P>
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<P>我还是再多看看这书，看它到底讲清楚了什么吧。</P>

Cielo

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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？

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<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>
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<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>