Editing Invariant (mathematics) (section)

== Formal statement ==
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The notion of invariance is formalized in three different ways in mathematics: via [[group action]]s, presentations, and deformation.

=== Unchanged under group action ===
Firstly, if one has a [[group (mathematics)|group]] ''G'' [[group action|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 ''[[coinvariant]]s,'' also known as ''orbits,'' which formalizes the notion of [[congruence relation|congruence]]: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the [[perimeter]] of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.

These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In [[classification problem (mathematics)|classification problem]]s, one might seek to find a [[complete set of invariants]], such that if two objects have the same values for this set of invariants, then they are congruent.

For example, triangles such that all three sides are equal are congruent under rigid motions, via [[Congruence (geometry)#Congruence of triangles|SSS congruence]], and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the [[Similarity (geometry)#Similar triangles|AAA similarity criterion]] shows that this is a complete set of invariants.

=== Independent of presentation ===
Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the [[Euler characteristic]] of a [[cell complex]] is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying [[topological space]] (the [[manifold]]) – as different cell complexes give the same underlying manifold, one may ask if the function is ''independent'' of choice of ''presentation,'' in which case it is an ''intrinsically'' defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.

The most common examples are:
* The [[Differentiable manifold#Definition|presentation of a manifold]] in terms of coordinate charts – invariants must be unchanged under [[change of coordinates]].
* Various [[manifold decomposition]]s, as discussed for Euler characteristic.
* Invariants of a [[presentation of a group]].

=== Unchanged under perturbation ===
Thirdly, if one is studying an object which varies in a family, as is common in [[algebraic geometry]] and [[differential geometry]], one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).