Fixed pointsActions of groupsOrbitsStabilizers


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 "at"
UNI-Graz Institut für Mathematik
UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

Fixed pointsActions of groupsOrbitsStabilizers