AgOrbitStabilizer( U, gens, omega )
AgOrbitStabilizer( U, gens, omega, f )
Let U be an ag group acting on a set Omega. Let omega be an
element of Omega. Then AgOrbitStabilizer returns the
point-stabilizer of omega in the group U and the orbit of
omega under this group. The stabilizer and orbit are returned as
record R with components R.stabilizer and R.orbit.
R.stabilizer is the point-stabilizer of omega. R.orbit is
the list of the elements of <omega> ^ <U>.
Let (u_1, ..., u_n) be an induced generating system of U and gens
be a list h_1, ..., h_n of generators of a group H, such that the map
u_imapsto h_i extends to an homomorphism alpha from U to H,
which is compatible with the action of G and H on Omega, such that
g in Stab_U( <omega> ) if and only if g^alpha in Stab_H( <omega>
). If f is missing OnRight is assumed, a typical application of
this function being the linear action of U on an vector space. If f
is OnPoints then ^ is used as operation of H on Omega.
Otherwise f must be a function, which takes two arguments, the first
one must be a point p of Omega and the second an element h of H
and which returns p ^ h.
gap> AgOrbitStabilizer( s4, [a,b,c,d], d, OnPoints );
rec(
stabilizer := Subgroup( s4, [ a, c, d ] ),
orbit := [ d, c*d, c ] )
GAP 3.4.4