### 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 _{G}X.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001