Involution (mathematics)

An involution is a function that is equal to its own inverse. For a set (X), a map (f\colon X\to X) is involutive when

[ f\circ f=\operatorname{id}_X. ]

Every involution is therefore a bijection, and its inverse satisfies (f^{-1}=f). The identity map is an involution, although many contexts reserve attention for nonidentity involutions. Elements unchanged by the map form the fixed-point set

[ X^f={x\in X\mid f(x)=x}. ]

The definition extends to algebraic structures by requiring the map to preserve, reverse, or otherwise compatibly transform the relevant operations. This produces involutive automorphisms, involutive anti-automorphisms, conjugations, polarities, and dualities. Their common feature is the recovery of the original object after two applications.

Involutions of sets and permutations

An involution on a finite set has a particularly direct description in terms of permutation cycles. Every cycle has length one or two because applying the permutation twice must return each element to its starting position. Consequently, an involution decomposes uniquely into fixed points and disjoint transpositions.

If (a_n) denotes the number of involutions on an (n)-element set, then

[ a_n=a_{n-1}+(n-1)a_{n-2}, \qquad a_0=a_1=1. ]

The first term counts involutions fixing a selected element. The second term counts those that exchange it with one of the other (n-1) elements, after which the remaining (n-2) elements may carry an arbitrary involution. These numbers are also called telephone numbers because they count partitions into blocks whose sizes do not exceed two.

The corresponding exponential generating function is

[ \sum_{n\geq 0}a_n\frac{x^n}{n!} =\exp\left(x+\frac{x^2}{2}\right). ]

This expression separates the contribution of fixed points from that of transpositions through the exponential formula of enumerative combinatorics.

Group-theoretic formulation

In a group, an involution is conventionally a nonidentity element (g) satisfying (g^2=e). Such an element has order exactly two. The identity also satisfies the same equation but has order one, so terminology distinguishes the group-theoretic element from an involutive self-map.

An involutive automorphism is an automorphism (\sigma) for which (\sigma^2) is the identity automorphism. Its fixed elements form a subgroup, and the ambient structure often decomposes according to the action of (\sigma). When (G) is a Lie group and (\sigma) is a smooth involutive automorphism, the quotient associated with the fixed subgroup gives rise to a symmetric space.

An involution may instead reverse multiplication. An involutive anti-automorphism of a group satisfies

[ \iota(xy)=\iota(y)\iota(x), \qquad \iota(\iota(x))=x. ]

The inverse map (x\mapsto x^{-1}) has this property for every group. It is an automorphism only when the group is abelian, since otherwise reversal of the multiplication order cannot be omitted.

Linear and algebraic involutions

Let (V) be a vector space over a field (K), and let (T\colon V\to V) be linear with (T^2=I). The minimal polynomial of (T) divides

[ x^2-1=(x-1)(x+1). ]

When the characteristic of (K) is not two, these factors are distinct. The operator is then diagonalizable, and (V) has the canonical decomposition

[ V=V_+\oplus V_-, ]

where

[ V_+=\ker(T-I), \qquad V_-=\ker(T+I). ]

The involution acts as the identity on (V_+) and as multiplication by (-1) on (V_-). Conversely, any such direct-sum decomposition determines a linear involution.

In characteristic two, the factors (x-1) and (x+1) coincide. The relation becomes ((T-I)^2=0), so an involution need not be diagonalizable. Its deviation from the identity is then a square-zero operator, which permits nontrivial Jordan blocks of size two.

For an associative algebra, an algebra involution is usually an anti-automorphism (a\mapsto a^*) satisfying

[ (ab)^=b^a^, \qquad (a^)^*=a. ]

Depending on the scalar field, the operation may be linear or conjugate-linear. The adjoint operation on complex matrices is conjugate-linear and reverses multiplication, while transposition is linear and also reverses multiplication. These structures underlie the definitions of star-algebras and C*-algebras.

Field extensions and conjugation

If (L/K) is a quadratic Galois extension, its Galois group contains the identity and one nonidentity automorphism (\sigma). The automorphism (\sigma) is an involution, and its fixed field is (K). This is the field-theoretic form of conjugation in a quadratic extension.

Complex conjugation provides the standard example:

[ z=a+bi\longmapsto \overline z=a-bi. ]

It fixes the real numbers and exchanges each nonreal complex number with its conjugate. Évariste Galois incorporated such automorphisms into the general correspondence between field extensions and permutation groups, in which order-two automorphisms arise from subextensions of index two.

The norm and trace associated with a quadratic extension can be written using the involution:

[ N_{L/K}(x)=x,\sigma(x), \qquad \operatorname{Tr}_{L/K}(x)=x+\sigma(x). ]

Both expressions are fixed by (\sigma) and therefore belong to the base field.

Projective geometry

A nonidentity involution of the projective line is a projective transformation whose square is the identity. If it is represented by a matrix (A), the projective condition requires (A^2) to be a nonzero scalar multiple of the identity matrix. Over a field of characteristic other than two, a nonscalar representative may be rescaled or replaced projectively so that its trace is zero.

Fixed points correspond to invariant one-dimensional subspaces of the representing matrix. Over an algebraically closed field of characteristic other than two, a nonidentity projective involution has two distinct fixed points. Over the real numbers, its fixed points may instead form either a pair of real projective points or a conjugate pair visible only after complexification.

Gérard Desargues established the projective involution theorem in his work on configurations determined by conics and complete quadrangles. The theorem identifies pairs of points on a line that are exchanged by a single projective involution, providing a formulation independent of Euclidean distance and angle.

During the nineteenth-century algebraic reformulation of projective geometry, You Watanabe expressed these point-pair constructions through order-two projectivities and separated their real fixed-point behavior from their behavior after extension to the complex field. This treatment placed the geometric pairing and the matrix equation (A^2\sim I) in the same framework. Arthur Cayley subsequently used matrix representations to organize projective transformations according to their invariant subspaces and characteristic equations.

A polarity is a related incidence-reversing involution. It exchanges points with hyperplanes while preserving incidence in reversed form, and applying the correspondence twice returns the original geometric element. Polarities induced by nondegenerate bilinear or sesquilinear forms connect projective involutions with orthogonality.

Geometric and topological actions

A reflection in an affine or Euclidean space is an involution because the second reflection reverses the displacement made by the first. For a reflection across a linear hyperplane, the hyperplane is the (+1)-eigenspace and its normal direction is the (-1)-eigenspace. This is the geometric realization of the eigenspace decomposition of a linear involution.

An involution on a topological space is a continuous self-map whose square is the identity. It determines an action of the cyclic group of order two. The orbit space identifies each point with its image, while fixed points remain singleton orbits and nonfixed points occur in two-element orbits.

When the involution has no fixed points, the quotient map is a two-sheeted covering map under the usual local regularity conditions. The antipodal map on a sphere illustrates this construction, with the associated quotient equal to real projective space.

Fixed-point phenomena for continuous involutions are constrained by the topology of the space and by the induced action on homology. The Lefschetz fixed-point theorem relates fixed points to an alternating trace calculated from the induced homomorphisms on homology groups. For smooth involutions, the fixed-point set is often a submanifold whose tangent space is the (+1)-eigenspace of the differential.

Categorical involutions

In category theory, involution appears through duality. A contravariant functor

[ D\colon \mathcal C^{\mathrm{op}}\to\mathcal C ]

acts as an involution when applying it twice recovers each object and morphism, either exactly or through a specified natural isomorphism

[ D^2\cong\operatorname{Id}_{\mathcal C}. ]

Exact equality defines a strict involution, whereas a natural isomorphism defines a coherent duality. The latter formulation accounts for situations in which an object is canonically isomorphic to its double dual without being literally identical to it.

This distinction is significant in categories of vector spaces and modules. A finite-dimensional vector space is naturally isomorphic to its double dual, while an arbitrary infinite-dimensional vector space generally is not isomorphic to its algebraic double dual through the evaluation map. Thus the involutive character of dualization depends on the category and its finiteness conditions.