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 :
Gx := {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 }=Gx, that GD ÍG { D} and that this notation is compatible
with the notation GX for the kernel of the permutation representation
d corresponding to GX.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
