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From: "Martin Schoenert"
To: Cube-Lovers@ai.mit.edu
Cc: BRYAN@wvnvm.wvnet.edu
In-Reply-To: "Jerry Bryan"'s message of Sun, 21 May 1995 07:18:49 -0400 (EDT)
Subject: Re: AntiSlice Under M-conjugacy (and a problem with slice)
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 of is the set of elements
that are G-conjugated to (i.e., there exists an element in ,
such that ^-1 * * = ).
Thus GAP is using -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 , then this had to be so,
because in this case every M-conjugacy class would be a subset of a
-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 -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