set under the mapping when Note that the elements of S are not fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space. An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. In linear algebra, if a linear transformation T has an eigenvector v, then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an invariant subspace which is stable under T. When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points. Formal statement The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation. Unchanged under group action Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group. Frequently one will have a group acting on a set X, which leaves one to determine which objects in an associated set F(X) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action. More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions. Dual to the notion of invariants are coinvariants, also known