[GAP Forum] orbits of a set under induced group action

[Laurent Bartholdi]

Hi Abhishek,

Use the command Orbits().

Functions X -> Y are (characteristic functions of) subsets of X. The orbits
split according to size.

For example, if G=Alt(4) acting on 4 points, try

gap> Orbits(AlternatingGroup(4),Combinations([1..4]),OnSets);
[ [ [  ] ], [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ],
  [ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ],
  [ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 2, 4 ], [ 2, 3, 4 ] ], [ [ 1, 2, 3, 4 ] ] ]

which tells you that there are five orbits, of sizes 1,4,6,4,1; for D_8, you get

gap> Orbits(DihedralGroup(IsPermGroup,8),Combinations([1..4]),OnSets);
[ [ [  ] ], [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ], [ [ 1, 2 ], [ 2, 3 ], [ 1,
4 ], [ 3, 4 ] ],
  [ [ 1, 2, 3 ], [ 2, 3, 4 ], [ 1, 3, 4 ], [ 1, 2, 4 ] ], [ [ 1, 2, 3, 4 ] ],
  [ [ 1, 3 ], [ 2, 4 ] ] ]

whence 6 orbits of size 1,4,4,2,4,1.

On 4/13/06, Abhishek <akshriv at gmail.com> wrote:
> Dear GAP forum,
>
> I need to solve the following problem.
>
> Let X be a set and G a group action on X. Let Y={0,1}.
> Consider the collection Y^X of all functions f:X --> Y.
> I need to find the orbits of Y^X induced by G. (The image of any function f
> in Y^X under a \in G is given by (a*f)(x)=f(a*x), for each x in X)
>
> I don't know if this can be solved using GAP since I have little background
> in Group Theory. I'll be glad if someone could point to some specific
> topic(s) in the GAP Manuals or beyond.
>
> Thank you,
> Abhishek
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Laurent Bartholdi          \  laurent.bartholdi<at>gmail<dot>com
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