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 is a trivial involution.

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.

Common examples in mathematics of 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.

See also: Star-algebra