The isotropy group, also called the stabilizer, of a group 
acting on a set 
at an element 
is the subgroup
of group elements that fix 
, defined by
 |
For example, consider the group SO(3) 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
give the usual group action of the unit
circle 
on the equator.
When two elements 
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 
for some element 
on the same orbit as 
.
