Some elements of a group G acting on a space X may fix a point x. These group elements form a subgroup called the isotropy group, defined by G_x = {g element G:g x = x}. For example, consider the group S O(3) of all rotations of a sphere S^2. Let x be the north pole (0, 0, 1). Then a rotation which does not change x 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 S^1 on the equator.

effective action | free action | group action | group orbit | group representation | Lie group quotient space | matrix group | topological group | transitive