I guess I am going to have to break down and get a copy of GAP. It
is truly impressive how much GAP can do so easily.
My interpretation of Martin's GAP program is that it implements the
general outline of the algorithm I described, except that GAP was
able to calculate the number of K-symmetric permutations in a very
simple and direct way, whereas I was going to have to puzzle each one
out by hand.
The heart of Martin's program appears to be the following, and I have
a couple of questions.
# compute how many elements have at least this symmetry group
number := Size( Centralizer( G, rep ) );
The first question is: how does the Size function work? As a simpler
example than the one above, what if you simply say Size(G)? I am
naively assuming that G is specified to GAP in terms of generators
only, and that it makes no attempt to actually represent each element
of G (too big!). And I have seen snippets of GAP libraries for G
posted by Mark Longridge, and they look like generators. I have been
in several group theory books lately, and I don't recall seeing
a general algorithm presented for determining the size of a finite
group based on its generators.
The second question is like unto the first: how does the
Centralizer function work? In this particular case, we don't
really need the Centralizer, we only need the Size of the Centralizer,
but the question remains in either case. Surely, GAP does not
literally try each element of G and each element of rep to see
which elements commute (too big again). So what is the general
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