Dear forum members:
Bruce W. Colletti asked:
>Am using GAP 3.4.4 and wonder how to find the orbits of the following group
>action, if it can be done at all.
>What are the distinct patterns that 9 beads form on a loop--3 black, 6 whit=
>The group is the 9th dihedral group (symmetries of a regular 9-gon) and the
>set acted upon would be the COMBINATION(9,3) 9-tuples of black/white
This is quite straightforward in GAP:
orbs :=3D Orbits (DihedralGroup (18), Combinations([1..9],3), OnSets);
gives you a list of the orbits in no time;
reps :=3D List (orbs, x -> x); gives you the following set of representatives [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 2, 6 ], [ 1, 3, 5 ], [ 1, 3, 6 ], [ 1, 4, 7 ] ]
Note that even if the way the group operates on a given domain is not
predefined (as is the operation "OnSets" in the above example), it is
possible to define the operation by an arbitrary GAP function (see "Other
Operations" in the GAP manual).
Hope this helps.