Group action in GAP

Question

I want to define a group action of group $G$ on set $X$, $G \times X \to X$ in GAP. I read the manual but I couldn't understand how. For example, I want to define group action of group $G$ on itself as $G \times G \to G$ as $(g,h) \to g^2hg^{-2} $. I would appreciate any help.

Answer 1

First, group actions in GAP are from the right. (No, this cannot be configured, is built deep in the system, and we have no reason to change this.) Thus you have an action $(h,g)\mapsto g^{-2}hg^2$.

Define a function implementing this action:

squareconj:=function(h,g)
  return h^(g^2);
end;

Now you can specify this action function as last argument of operations, such as Orbit or OrbitsDomain:

gap> G:=SymmetricGroup(5);;
gap> o:=OrbitsDomain(G,G,squareconj);;
gap> List(o,Length);
[ 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 5, 5, 5, 5, 5, 1,
  5, 1, 1 ]
gap> o[3];
[ (3,4,5), (1,2,5), (2,3,4), (1,4,5), (1,2,3) ]
gap> s:=Stabilizer(G,(1,5)(2,4),squareconj);
Group([ (1,2), (1,5), (4,5) ])
gap> RepresentativeAction(G,(3,4,5),(1,4,5),squareconj);
(1,4,2,5,3)