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 .