Involution

In mathematics, an involution is a function that is its own inverse, so that

f(f(x)) = x for all x in the domain of f.

The identity map provides a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, reflections in geometry, complementation in set theory and complex conjugation. The P-symmetry in physics is a deep application of the idea.

A famous geometric involution is the inversion, that is a mapping of the plane into itself, which exchanges the interior and the exterior of a circle and takes the role in inversive geometry of the reflection in Euclidean geometry.

Other examples include include the ROT13 transformation and the Enigma cipher.

In group theory, an element of a group is an involution if it has order 2; i.e., if a is an element of the group and i the identity element, a is an involution if <math> a \cdot a = i <math>. For example, a permutation is an involution if it a product of non-overlapping transpositions.

The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups.

Coxeter groups are groups generated by their involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.

In ring theory, an involution is an antihomomorphism whose square is the identity. Examples of involutions include complex conjugation and the transpose of a matrix.

See also: Star-algebra

fr:Involution ja:対合