   Date: Fri, 03 Mar 95 02:38:00 -0500   From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Centralizers & GAP

Martin wrote:

```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).
```

Ok.... let's see if I understand this "centralizer" business

```(2 ^ 12 / 2 ) * 12! =  980,995,276,800 elements in GE
G is the Group of the cube
GE is the Group of the Edges Only cube
P is the element we call the Pons Asinorum (or 6 X order 2)
```

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

That is, out of all the elements of GE (or G) only 2,949,120 of them
commute with the pons asinorum.

Let's represent the Group of elements of GE that commute with P
as X. Elements of X are represented by x.

Then in all x of X, xP = Px.

But what is really troubling me is:

* How do you represent a particular cube position (e.g. pons)
with GAP? *

If I could do that, then I could verify how many elements of
the cube group commute with a given cube position:

Size (Centralizer (cube, pons));

```Should give 2949120 (2,949,120) ... right Martin?
```

and

Size (Centralizer (sq, cube.centre));
663552

```-> Mark <-
```     