Date: Mon, 06 Mar 95 23:44:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ Subject: New GAP insights

Okay, I understand the GAP conventions better now.

If we adhere to the following model for the cube:

```               +--------------+
|  1    2    3 |
|  4  top    5 |
|  6    7    8 |
+--------------+--------------+--------------+--------------+
|  9   10   11 | 17   18   19 | 25   26   27 | 33   34   35 |
| 12  left  13 | 20 front  21 | 28 right  29 | 36  rear  37 |
| 14   15   16 | 22   23   24 | 30   31   32 | 38   39   40 |
+--------------+--------------+--------------+--------------+
| 41   42   43 |
| 44 bottom 45 |
| 46   47   48 |
+--------------+
```
```Then the Pons Asinorum would be:
pons := ( 2,42)( 4,45)( 5,44)( 7,47)(10,31)(12,28)(13,29)(15,26)
(18,39)(20,36)(21,37)(23,34);;

And the slice group would be:
slice := Group(
( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)
(41,46,48,43)(42,44,47,45)(14,38,30,22)(15,39,31,23)(16,40,32,24),
( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)
(25,30,32,27)(26,28,31,29)( 3,19,43,38)( 5,21,45,36)( 8,24,48,33),
(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)
(33,38,40,35)(34,36,39,37)( 3,32,46, 9)( 2,29,47,12)( 1,27,48,14)
);;

And the anti-slice group would be:
antisl := Group(
( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)
(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40),
( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)
(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)
(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)
);;

Size (antisl) = 6,144
Size (slice)  =   768
```

These numbers concur with Mr. Singmaster's earlier "Notes".

The following command shows that pons is at the centre of slice group:
Size (Centralizer (slice, pons)) = 768

Once again, I will refer to Martin's earlier statement about
centralizers:

```That is, of the total 980995276800 elements in GE
only 980995276800/332640 = 2949120 elements centralize P.
And I used the definition of P from your e-mail of 1995/01/03,
i.e., P = (F2 B2) (U2 D2) (L2 R2) = (F2 B2) (L2 R2) (U2 D2) = ...
(one gets the same element independent of the order of the
three pairs).
```

So now that I have the groups and pons element correct:
Size (Centralizer (edge, pons)) = 2,949,120

I wrote some statements before....

Only 2,949,120 elements of GE centralize P,
also only...
2,949,120 elements of G centralize P

I am only partly correct as....

```Size (Centralizer (cube, pons)) = 130,026,464,870,400
```

As Martin said before:
> Only one out of 332640 elements of GE (and of G) centralizes P.

```Size (cube) / 332640 = 130,026,464,870,400
or 130 trillion and change.
```

...the full cube group has many more elements which commute with
pons than the mere edge group!

GAP is a very function-laden beastie:

Size (Intersection (antisl, slice)) = 8
This function gives the number of elements included in both the
anti-slice and slice groups.

Naturally there is a corresponding Union function.

Since I have studied the squares group and the <U, R> group, the
number of elements in the intersection of the two are of
particular interest:

```Size (Intersection (ur, sq)) = 72

And now we have a new way to check an old result  :-)
Order (cube, uturn * rturn) = 105
```

Of course, now that I have answered my old questions, I must
formulate new ones....

A) What is the next most commutative element (the pancentre?)
after the 12-flip?
B) What is the least commutative element (the anticentre?) of
the cube group?

```-> Mark <-
```