[GAP Forum] building group action without losing original names?

[Alexander Hulpke]

Dear Forum,

On Dec 19, 2012, at 9:30 AM, Frederic Vanhove <fvanhove at cage.UGent.be> wrote:
> 
> 
> I have yet another question on (equivalent) group actions.
> When I build an action on a set, is there way to have GAP store a bijection between the original points on which he acts, and the labels [1..v]?

The bijection is always simple [1..v], i.e. the action is on the elements as arranged in a listi, if neccessary calling `Enumerator' on the domain. In your example this correspondence is easier to see if you explicitly take an element list for G:

gap> elm:=AsSSortedList(G);
[ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), 
  (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), 
  (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), 
  (1,4)(2,3) ]
gap> act:=Action(g,G,OnPoints);
<permutation group with 4 generators>
gap> Orbits(act);
[ [ 2, 6, 7, 3, 15, 22 ], [ 4, 5, 12, 21, 20, 16, 13, 9 ], [ 8, 17, 24 ], 
  [ 10, 14, 11, 19, 23, 18 ] ]
gap> elm{ [2, 6, 7, 3, 15, 22 ]}; # one orbit
[ (3,4), (2,4), (1,2), (2,3), (1,3), (1,4) ]

(This is specified in the manual section
41.7 The Permutation Image of an Action
http://www.gap-system.org/Manuals/doc/ref/chap41.html#X87F73CCA7921DE65

Best wishes,

    Alexander Hulpke

-- Colorado State University, Department of Mathematics,
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email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke