Some elements of a group 
acting on a space 
may fix a point 
. These group elements form a subgroup
called the isotropy group, defined by
 |
For example, consider the group 
of all rotations of a sphere 
. Let 
be the north pole 
. Then a rotation which does not change 
must turn about the usual axis, leaving the north
pole and the south pole fixed. These rotations
correspond to the action of the circle group 
on the equator.
When two points 
and 
are on the same group orbit, say 
, then the isotropy groups are conjugate
subgroups. More precisely, 
. In fact, any subgroup conjugate to 
occurs as an isotropy group 
to some point 
on the same orbit as 
.
