In mathematics, an involution is a function that is its own inverse, so that Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

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. In mathematics, the domain of a function is the set of all input values to the function. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In its simplest form, multiplication is a quick way of adding identical numbers. ... Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym for number theory. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... The word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Thales (circa 624-547 BC) dealing with spatial relationships. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... P-symmetry is simply the spatial symmetry exhibited during a reflection. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ...

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. In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... In mathematics, a plane is the fundamental two-dimensional object. ... In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

Other examples include include the ROT13 transformation, the Beaufort polyalphabetic cipher, and the Enigma cipher. ROT13 replaces each letter by its partner 13 characters further along the alphabet. ... Beaufort is: The name of some places in the United States of America: Beaufort, North Carolina Beaufort, South Carolina Beaufort County, North Carolina Beaufort County, South Carolina The name of several communes in France: Beaufort, in the Haute-Garonne département Beaufort, in the Hérault département Beaufort, in the Isère département... A polyalphabetic cipher is any cipher based on substitution, using multiple substitution alphabets. ... In the history of cryptography, the Enigma was a portable cipher machine used to encrypt and decrypt secret messages. ... This article is about algorithms for encryption and decryption. ...

An involution is a kind of bijection. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...

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. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... See transposition for meanings of this term in telecommunication and music. ...

See also star-algebra. In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear antiautomorphism * : A->A which is an involution. ...

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 = 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. Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

A permutation is an involution precisely if it can be written as a product of non-overlapping transpositions. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

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. The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the 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 mathematics, a Coxeter group is a group with a presentation of the form where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being regular polygons of the same size and shape, and the same number of faces meeting at each of its vertices. ... A dodecahedron, one of the five Platonic solids. ...