The functions MovedPoints, NrMovedPoints, LargestMovedPoint,
and SmallestMovedPoint are defined for arbitrary collections of
permutations (see Moved Points of Permutations),
in particular they can be applied to permutation groups.
gap> g:= Group( (2,3,5,6), (2,3) );; gap> MovedPoints( g ); NrMovedPoints( g ); [ 2, 3, 5, 6 ] 4 gap> LargestMovedPoint( g ); SmallestMovedPoint( g ); 6 2
The action of a permutation group on the positive integers is a group
action (via the acting function OnPoints).
Therefore all action functions can be applied
(see the Chapter Group Actions),
for example Orbit, Stabilizer, Blocks, IsTransitive, IsPrimitive.
If one has a list of group generators and is interested in the moved points
(see above) or orbits, it may be useful to avoid the explicit construction
of the group for efficiency reasons.
For the special case of the action of permutations on positive integers
via ^, the following functions are provided for this purpose.
OrbitPerms( perms, pnt ) F
returns the orbit of the positive integer pnt under the group generated by the permutations in the list perms.
OrbitsPerms( perms, D ) F
returns the list of orbits of the positive integers in the list D under the group generated by the permutations in the list perms.
gap> OrbitPerms( [ (1,2,3)(4,5), (3,6) ], 1 ); [ 1, 2, 3, 6 ] gap> OrbitsPerms( [ (1,2,3)(4,5), (3,6) ], [ 1 .. 6 ] ); [ [ 1, 2, 3, 6 ], [ 4, 5 ] ]
Similarly, several functions concerning the natural action of
permutation groups address stabilizer chains (see Stabilizer Chains)
rather than permutation groups themselves, for example BaseStabChain
(see BaseStabChain).
GAP 4 manual