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Sorry, lately I didn't have any time to follow the discussions on

Cube-Lovers (preparing the second upgrade of GAP 3.4 and working long

hours for GAP 4.0). But Jerry Bryan's message talks about GAP's

'NumberConjugacyClasses' function.

Jerry wrote

I doubt that Mark's theory about GAP using C-conjugacy for slice

instead of M-conjugacy is correct. I already have 50 positions

to 23 for GAP, and using C-conjugacy would just make my results

larger. For example, RL' and R'L are M-conjugate positions,

but not C-conjugate positions. I don't have a clue why my

results do not match GAP. I have double and triple checked

my results, and they seem correct. For example, I can "expand"

my conjugacy classes, and the results then match Mark's exactly.

GAP's 'NumberConjugacyClasses' follows the general usage in group theory.

The conjugacy class of an element <g> of <G> is the set of elements <h>

that are G-conjugated to <g> (i.e., there exists an element <x> in <G>,

such that <x>^-1 * <g> * <x> = <h>).

Thus GAP is using <slice>-conjugacy classes. Since GAP is using a

*larger* group, it is not surprising that GAP finds less conjugacy

classes (if M were a subgroup of <slice>, then this had to be so,

because in this case every M-conjugacy class would be a subset of a

<slice>-conjugacy class).

Jerry continued

How does GAP's NumberConjugacyClasses function work? By that,

I mean how does it know the subgroup with respect to which

you are taking conjugacy classes (if my terminology is correct)?

For example, how does it know to take C or M or whatever

conjugates?

It always takes the whole group itself as the acting group. With some

related functions (e.g., 'ConjugacyClass' itself) you can specify that

you want another group acting, but not with 'NumberConjugacyClasses'.

Mark Longridge wrote (as cited by Jerry)

In your calculations of M-conjugacy classes for the slice group the

total number of classes is 50, but I think GAP does not use

M-conjugates but C-conjugates instead. The NumberConjugacyClasses

function always thrashes with any larger groups unfortunately.

If you could easily tweak your program perhaps you could

verify my theory.

As I wrote above GAP does not use C-conjugates but <slice>-conjugates.

Conjugacy classes in permutation groups are notoriously difficult.

Computing the conjugacy classes of G (the full cube group) for example

is absolutely impossible (without using some theory anyway).

Martin.

-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany