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

[Petra Holmes]

You can look at each orbit of G on the points separately.  To do this, you
can use the code:

o:=Orbits(G); ims:=[];
for oo in o do
	hom:=ActionHomomorphism(G,oo,OnPoints);
	Add(ims,Image(hom));
od;

Then you can look at blocks in each orbit of G separately, as you are now
dealing with a collection of transitive groups.

Beth Holmes

On Tue, 30 Nov 2004 wh at icparc.ic.ac.uk wrote:

> 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.
> 
> Thanks,
> Warwick
> 
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