[GAP Forum] Partitions and orbits of automorphism groups

[Leonard Soicher]

Dear GAP-Forum,

Note that you can use the GRAPE function AutGroupGraph (which calls nauty)
to determine directly the stabiliser of an ordered partition of the vertices 
of a graph in the automorphism group of that graph. See the documentation
for AutGroupGraph. For example:

gap> LoadPackage("grape");
true
gap> J:=JohnsonGraph(4,2);
rec( adjacencies := [ [ 2, 3, 4, 5 ] ], group := Group([ (1,4,6,3)(2,5), (2,4)(3,5) ]), isGraph := true,
  isSimple := true, names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ], order := 6,
  representatives := [ 1 ], schreierVector := [ -1, 2, 1, 1, 1, 1 ] )
gap> H:=AutGroupGraph(J,[[1],[6],[2,5],[3,4]]);
Group([ (3,4), (2,5) ])
gap> Size(H);
4
gap> Orbits(H,[1..J.order]);
[ [ 1 ], [ 2, 5 ], [ 3, 4 ], [ 6 ] ]


Regards,
Leonard
 
----- Original Message -----
From: "Sven Reichard" <Sven.Reichard at tu-dresden.de>
To: forum at gap-system.org
Sent: Thursday, 11 July, 2013 8:54:07 AM
Subject: Re: [GAP Forum] Partitions and orbits of automorphism groups

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Am 11.07.2013 09:43, schrieb Matan Ziv-Av:
> On Thu, 11 Jul 2013, Ebrahim Ghorbani wrote:
> 
>> Dear Forum,
>> 
>> I have a partition of the vertex set of a vertex-transitive graph
>> G. I guess that this partition are the orbit partition of some
>> subgroup of Aut(G). Is there any way to find out this?
> 
> Find the stabilizer (as a tuple of sets) of the partition in
> Aut(G). If the orbits of any subgroup gives this partition, then
> the stabilizer also does.
> 
> 

Of course Matan is right; we need to consider tuples of sets, not sets
of sets as I said earlier.

Sven.

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