[GAP Forum] Can I safely identify the moved points of PGL(n, 2) with GF(2)^n\0?

[Max Neunhoeffer]

Dear Gordon,

as far as I know the documentation does not guarantee this. However,
here is a way to guarantee some identification using the orb package
(here for n=8 by way of example):

gap> g := GL(IsMatrixGroup,8,2);
SL(8,2)
gap> v := ShallowCopy(Zero(GF(2)^8));
<a GF2 vector of length 8>
gap> v[1] := Z(2);
Z(2)^0
gap> o := Orb(g,v,OnLines,rec( storenumbers := true ));
<open orbit, 1 points>
gap> Enumerate(o);
<closed orbit, 255 points>
gap> permgens := ActionOnOrbit(o,GeneratorsOfGroup(g));;

It does not give you exactly the numbering you want but on the other hand

  o[i]

gives you quickly the i-th vector and

  Position(o,v)

for a vector v uses a hash table and thus gives you the number of a vector
in constant time.

Best regards,
  Max

On Wed, Mar 13, 2013 at 03:52:30PM +0800, Gordon Royle wrote:
> If I use the groups
> 
> PGL(n,2)
> 
> in GAP, then is it guaranteed that the set permuted by this group can be identified in the obvious fashion with GF(2)^n \ 0?
> 
> What I mean is that it would be natural to assign the non-zero vectors in GF(2)^n\0 to integers in the range {1,2,…,2^n-1} simply by treating a 0/1 vector as the binary representation of an integer.
> 
> 0000001 -> 1
> 0000010 -> 2
> 0000011 -> 3
> 
> and so on.
> 
> >>From my experiments, GAP *appears* to behave consistently with this (for example, if I check the stabiliser of [x, y, x ^ y] where x ^ y is bitwise binary XOR, then it is always the correct thing).
> 
> But can I rely on it?
> 
> 
> 
> 
> Professor Gordon Royle
> School of Mathematics and Statistics
> University of Western Australia
> Gordon.Royle at uwa.edu.au<mailto:Gordon.Royle at uwa.edu.au>
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
> 
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-- 
Max Neunhoeffer              http://www-groups.mcs.st-and.ac.uk/~neunhoef/
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