Involuties

Involuties, also called involutions, are mathematical objects that are their own inverse: applying them twice yields the identity. They occur in various areas, including functions, linear algebra, and group theory, often modeling a kind of symmetry.

In the functional sense, an involution is a map f from a set to itself such that f(f(x)) = x for every x in the domain. Examples include the identity map, negation on a vector space (f(v) = -v), and geometric reflections across a line or plane. Involutions are always bijective, since they have inverses equal to themselves.

In linear algebra, an involutive linear operator T satisfies T^2 = I, where I is the identity operator. Such operators have eigenvalues ±1, and over fields with characteristic not equal to 2 they are diagonalizable. A common example is a reflection operator, which fixes a subspace pointwise and negates its orthogonal complement.

In group theory, an involution is an element g of a group with g^2 = e, the group identity. In the symmetric group S_n, involutions correspond to permutations whose cycle decomposition consists only of fixed points and disjoint transpositions. They arise in the study of symmetries and in counting problems, where formulas quantify the number of involutions for a given n.

In ring theory, an involution refers to an involutive anti-automorphism: a map that reverses the order of multiplication and squares to the identity, such as complex conjugation on the complex numbers or adjoint operations in algebras.

Involutions provide a convenient framework for describing reflections, symmetries, and self-inverse structures across mathematics. See also involution (mathematics).