This page is about involutions in mathematics; for involution in philosophy and integral theory, see involution (philosophy).

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 is 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, the Beaufort polyalphabetic cipher, and the Enigma cipher.

An involution is a kind of bijection.

Involutions in ring theory

In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples include complex conjugation and the transpose of a matrix.

See also star-algebra.

Involutions in group theory

In group theory, an involution is an element of a group that has order 2; i.e. an element a such that a² = i = the identity element. Originally, this definition differed not at all from the first definition above, since members of groups were always bijections from a set into itself, i.e., group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution.

A permutation is an involution precisely if it can be written as 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.