[GAP Forum] A question about Action

[Stephen Linton]

> On 13 Feb 2016, at 20:24, Peter Cameron <pjc at mcs.st-andrews.ac.uk> wrote:
> 
> Dear GAP forum,
> 
> It seems that GAP can find representatives for all systems of minimal blocks
> of imprimitivity for an arbitrary group action, but all blocks only for a
> permutation group (and there seems no easy way to get all maximal blocks).
> 

You can build up from minimal blocks recursively, of course. Take the action on the blocks of a minimal block
system and find the minimal blocks there, they correspond to non-minimal block systems for the original action, etc.

> I have tried to get around this as follows. G is a group acting on O (which
> is actually an orbit on 2-subsets of [1..n]). I do the following:
> 
> GG:=Action(G,O,OnSets);
> BB:=AllBlocks(GG);
> B:=List(BB,x->List(x,y->O[y]));
> 
> This assumes that the Action maps each element of O to its position in the
> list, and I can't find a guarantee to this effect in the manual. Can I rely
> on this?

I believe so, although this is (at least) a documentation failing). The manual does say:

 40.3-2 Action homomorphisms
  
  See ActionHomomorphism (41.7-1).
  
  The  calculation of images is determined by the acting function used and -for large domains- is often dominated by the search for the position of an
  image in a list of the domain elements. This can be improved by sorting this list if an efficient method for \< (31.11-1) to compare elements of the
  domain is available.

which strongly suggests to me that it is retaining the given order for the domain.

	Steve


> 
> Thanks for any advice.
> 
> Peter Cameron.
> 
> 
> 
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