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

harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Stabilizers |