[GAP Forum] Blocks and block systems for non-transitive groups

wh at icparc.ic.ac.uk wh at icparc.ic.ac.uk
Tue Nov 30 18:54:11 GMT 2004

Dear GAP Forum,

I'm wanting to write some code that deals with blocks and block systems that
needs to work even if the group is not transitive.  Most of the GAP
functions I found in the documentation include things like "The action must
be transitive." or "... for a permutation group G acting transitively ...".
(Incidentally, MaximalBlocks doesn't specify it but does require it.)

That said, _some_ of these functions appear to give the right answer with
_some_ of the non-transitive groups I've tried - but not others.  :)

Being a relative novice when it comes to group theory, I was wondering
whether the restriction was because:

1)  It doesn't make sense to talk about blocks and block systems for
    non-transitive groups for some reason I haven't thought of.

2)  The standard algorithms only work for transitive groups.

3)  Nobody bothered to handle non-transitive groups because nobody cares
    about them in this context (except me :).

Actually, I suspect it is 3) combined with:

4)  If you assume transitivity there are some optimisations that can be
    used to speed up the computation.

Anyway, assuming it's not 1), does anybody have any GAP code that returns
"interesting" blocks (I'm mostly interested in large blocks, but not
exclusively) that works for non-transitive groups?  An analogue to AllBlocks
or similar would be great but perhaps too much to hope for.  :)  Or some
pointers on where to start if I want to write my own routines?  Or what
algorithms to avoid because they only work for transitive groups?

Any help much appreciated.


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