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

[wh at icparc.ic.ac.uk]

Dear Beth, Dear GAP Forum,

On Wed, Dec 01, 2004 at 03:53:25PM +0000, Petra Holmes wrote:
> 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.

Thank you.  That would indeed allow me to find all the blocks I wanted
within each orbit using standard routines.

Having given some more thought to the problem, however, I am particularly
interested in blocks than span more than one orbit for intransitive groups.
To give an example, the group generated by (1,2,3,4)(5,6,7,8) is
intransitive: it has two orbits.  Considering the orbits independently one
obtains the representative blocks [1,3] and [5,7].  But there are also block
systems with representative blocks [1,3,5,7] and [1,5] that I would like to
find but cannot by considering the orbits independently.  Perhaps they can
be recovered/extracted later?  But unless that's straightforward to do it
seems as though one might as well use a block-finding algorithm that works
directly with intransitive groups (unless that turns out to be unexpectedly
hard).

I guess I should go study the source code for AllBlocks some more and see if
I can figure out how it works.  :)

Cheers,
Warwick