GAP4, Version: 4.dev of today, i686-pc-linux-gnu-gcc
Components:  small 2.0, small2 2.0, small3 2.0, small4 1.0, small5 1.0,
             small6 1.0, small7 1.0, small8 1.0, id2 3.0, id3 2.1, id4 1.0,
             id5 1.0, id6 1.0, trans 1.0, prim 2.1  loaded.
Packages:    AClib 1.1, Polycyclic 1.1, Alnuth 2.0, CrystCat 1.1.1,
             Cryst 4.1.3, AutPGrp 1.2, CRISP 1.2.1, CTblLib 1.1.3,
             TomLib 1.1.2, FactInt 1.3.1, GAPDoc 0.9999, LAGUNA 3.3,
             Sophus 1.12, Polenta 1.1, ResClasses 2.0.0, GUAVA 2.0  loaded.
gap> a:=2^6*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)*Binomial(6,2)*Binomial(4,2)/Factorial(6);
665280
gap> b:=2^6*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)*Binomial(6,2)*Binomial(4,2)/Factorial(5);
3991680
gap> c:=2^7*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)*Binomial(6,2)/Factorial(4);
6652800
gap> d:=2^8*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)/Factorial(3);
3548160
gap> e:=2^9*Binomial(12,2)*Binomial(10,2)/Factorial(2);
760320
gap> f:=2^10*Binomial(12,2);
67584
gap> g:=2^11 -1;
2047
gap> a+b+c+d+e+f+g;
15687871
gap> a+c+e+g;
8080447
gap> b+d+f;
7607424
gap>
gap>
gap> a0:=2^6*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)*Binomial(6,2)*Binomial(4,2)/Factorial(6);
665280
gap> b0:=2^7*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)*Binomial(6,2)*Binomial(4,2)/Factorial(5);
7983360
gap> c0:=2^8*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)*Binomial(6,2)/Factorial(4);
13305600
gap> d0:=2^9*Binomial(12,2)*Binomial(10,2)*Binomial(8,2)/Factorial(3);
7096320
gap> e0:=2^10*Binomial(12,2)*Binomial(10,2)/Factorial(2);
1520640
gap> f0:=2^11*Binomial(12,2);
135168
gap> a0+b0+c0+d0+e0+f0;
30706368
gap> case_a:=10395;
10395
gap> case_b:=a+c+e+g;
8080447
gap> case_c:=case_a*case_b;
83996246565
gap> case_d:=11424*(b+d+f);
86907211776
gap> num_order_2:=case_a+case_b+case_c+case_d;
170911549183
gap> case_a0:=10395;
10395
gap> case_b0:=a0+c0+e0;
15491520
gap> case_c0:=case_a0*case_b0;
161034350400
gap> case_d0:=11424*(b0+d0+f0);
173814423552
gap> num_order_2_old:=case_a0+case_b0+case_c0+case_d0;
334864275867
gap>
gap>


++++++++++++++++++++++++++++++++++++++++++++++







Subject:
Re: moves of order 2
From:
David Joyner <wdj@usna.edu>
Date:
Fri, 20 May 2005 05:26:27 -0400
To:
kociemba@t-online.de
CC:
David Joyner <jeeves-debian@comcast.net>, David Joyner <wdj@usna.edu>

Herbert Kociemba wrote:

> Am Thu, 19 May 2005 14:10:45 +0200 hat David Joyner <wdj@USNA.Navy.Mil>  geschrieben:
>
>> Hi Herbert:
>>
>> What is T? How do you define the extended group? Using
>> semidirect products?
>>
>
> Hello David,
>
> I am not very firm in group theory (want to improve my knowledge with your  book ;-)), but I hope the following is correct.
>
> Let M be the symmetry group of the cube with 48 elements and let N be the  group {e,T) containing the identity element e and the "inversion operator"  T.
> Then the extended group I talk from is the direct product MxN of these two  groups. With s in M, instead of (s,e) we write simply s and instead of  (s,T) we write T.s .


It seems to me that this is not a direct product. Probably it is a
semi-direct product though.

>
> Now we define a map R from MxN -> S(G), where G is the set of all cube  positions and S(G) the corresponding permutation group by
>
> R(s)(g) = sgs' and R(T.s)(g) = sg's'
>
>
> This is an homomorphism (if for a,b from S(G) we define a*b(g) = a(b(g)).


This property of R must be *proven* not defined. Indeed, both sides of
the equation are defined already, but what isn't clear is whether they are equal or not.

In fact, I am suspicious that the map R is a homomorphism (it is a bit trickier when the group involved is a semi-direct product). I'll ask a friend who knows group theory extremely well before saying anything
definite.



> So the Polya-Burnside-Lemma may be used to count the number of orbits,  which is the number of equivalence classes of cubes in the sense I want.


If R is not a homomorphism then I don't see how the P-B-L applies.

However, T.id certainly is a function (not a homomorphism) from the cube
group to itself and its number of fixed points is equal to one minus
the set of points of order 2.


>
> Now I have to count all fixpoints of the permutations of R(MxN). Dan Hoey  did this for the 48 elements s in M, so only the fixpoints of the 48  permutations of the form R(T.s) remain, that is we have to find for all s  of M the elements g of G with sg's=g, or equivalent sgs'=g'.
>
> For s=id we just search cubes g with g=g' and this are -as I think - what  you define as moves of order 2 (ignoring the id-Cube of course).
>
> And this are 10396*8080448 + 11424*7607424 cubes, if we take the even and  odd permutations of corners of edges. These numbers I computed in just the  same way you did it by hand (except of course what I think is not correct  in your computation) and double checked them with my cube explorer program  and with the numbers Mike Godfrey got with his program.
>
> Please have a critical view on the lines I wrote above and tell me, if  something is not right.
>
> Best wishes
>
> Herbert
>
>