38.4 Stabilizers

computes the orbit and the stabilizer of pnt simultaneously in a single Orbit-Stabilizer algorithm.

The stabilizer must have G as its parent.

computes the stabilizer in G of the point pnt, that is the subgroup of those elements of G that fix pnt. The stabilizer will have G as its parent.

gap> g:=Group((1,3,2),(2,4,3));;
gap> Stabilizer(g,4);
Group([ (1,3,2) ])

The stabilizer of a set or tuple of points can be computed by specifying an action of sets or tuples of points.

gap> Stabilizer(g,[1,2],OnSets);
Group( [ (1,2)(3,4) ] )
gap> Stabilizer(g,[1,2],OnTuples);
Group(())
gap> OrbitStabilizer(g,[1,2],OnSets);    
rec(
  orbit := [ [ 1, 2 ], [ 3, 4 ], [ 1, 3 ], [ 2, 4 ], [ 2, 3 ], [ 1, 4 ] ],
  stabilizer := Group( [ (1,2)(3,4) ] ) )

The standard methods for all these actions are an Orbit-Stabilizer algorithm. For permutation groups backtrack algorithms are used. For solvable groups an orbit-stabilizer algorithm for solvable groups, which uses the fact that the orbits of a normal subgroup form a block system (see SOGOS) is used.

This operation should not be called by a user. It is documented however for purposes to extend or maintain the group actions package.

OrbitStabilizerAlgorithm performs an orbit stabilizer algorithm for the group G acting with the generators gens via the generator images gens and the group action act on the element pnt. (For technical reasons pnt and act are put in one record with components pnt and act respectively.)

The pntact record may carry a component stabsub. If given, this must be a subgroup stabilizing all points in the domain and can be used to abbreviate stabilizer calculations.

The argument Omega (which may be replaced by false to be ignored) is the set within which the orbit is computed (once the orbit is the full domain, the orbit calculation may stop). If blist is given it must be a bit list corresponding to Omega in which elements which have been found already will be ``ticked off'' with true. (In particular, the entries for the orbit of pnt still must be all set to false). Again the remaining action domain (the bits set initially to false) can be used to stop if the orbit cannot grow any longer. Another use of the bit list is if Omega is an enumerator which can determine PositionCanonicals very quickly. In this situation it can be worth to search images not in the orbit found so far, but via their position in Omega and use a the bit list to keep track whether the element is in the orbit found so far.

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