Stabilizers

Stabilizers

To the orbits G(x), which are subsets of X, there correspond certain subgroups of G. For each x ÎX we introduce its stabilizer :

G_x := {g | gx=x }.

This is the subgroup of elements of G that stabilize the point x. There is also the subgroup of elements which pointwise stabilize the elements of a subset D of X:

G _D:= {g | " x ÎD:gx=x },

and which is called the pointwise stabilizer  

of D, in contrast to the setwise stabilizer  

of D:

G _{{ D}}:= {g | {gx | x ÎD}= D}.

We note in passing that G _{{x }}=G_x, that G_D ÍG _{{ D}} and that this notation is compatible with the notation G_X for the kernel of the permutation representation d corresponding to _GX.

Stabilizers