# Involution

*See involution (philosophy) for the philosophy meaning.*

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

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

## Contents

## General properties

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, complementation in set theory and complex conjugation.

Other examples include include circle inversion, the ROT13 transformation, and the Beaufort polyalphabetic cipher.

An involution is a kind of bijection.

## Involutions in Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Doing a reflection twice, brings us back where we started.

This transformation is a particular case of an affine involution.

## Involutions in differential geometry

In differential geometry, an involutive distribution is a certain type of subbundle of a vector bundle. According to Frobenius theorem, involutive distributions are completely integrable. They often used to generate a foliation.

## 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 element of a group is an involution if it has order 2; i.e. an involution is an element *a* such that *a*^{2} = *e*, where *e* is 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*. The group of bijections generated by an involution through composition, is isomorphic with cyclic group *C _{2}.*

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.