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In mathematics, an involution is a function that is its own inverse, so that
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 theoryAbstract algebra Group theory Group theory is that branch of mathematics concerned with the study of groups. Please refer to the Glossary of group theory for the definitions of terms used throughout group theory. See also list of group theory topics., an element of a groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G is an involution if it has orderIn group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. the number of its elements; the order of an element a of a group is the smallest positive integer m such that a m e (where e denotes the ide 2; i.e., if a is an element of the group and i the identity element, a is an involution if . For example, a permutationThis article is about permutation a mathematical concept. See permutation (music) for the application of this concept to music. In mathematics, the concept of a permutation expresses the idea that objects that can be distinguished may be arranged in vario is an involution if it a product of non-overlapping transpositionIn music, transposition is moving a note or collection of notes up or down in pitch by a constant interval. In mathematics, a transposition is a permutation which exchanges two elements and keeps all others fixed. For example (1 3) is a transposition whics.
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 groupsThe classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which classifies all of the finite simple groups. In all, the work comprises about 10,000 15,000 pages in 500 journal arti.
Coxeter groupGroup theory In mathematics, a Coxeter group is a group with a presentation of the form : where m ≥ 2; the condition m ∞ means no relation of the form x x ''m should be imposed. It is convenient to regard m as a symmetric function of the indicess 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.