involution

An **involution** is an endomorphism whose square is the identity morphism. Such an endomorphism must be an automorphism; indeed, it is its own inverse.

Where this makes sense, an **anti-involution** is an antihomomorphism instead of a homomorphism (so an antiendomorphism and necessarily an antiautomorphism).

Two involutions $f, g : X \to X$ commute if and only if their composition $f g$ is also an involution, as displayed by the following algebra:

$f g = f (f g f g ) g = (f f) g f (g g) = g f$

$(f g)(f g) = f (g f) g = f (f g) g = (f f)(g g)= 1$

In combinatorics, an important class of involutions are the fixed point free ones. The number of fixed point free involutions $f : X \to X$ on a finite set of even cardinality $|X| = 2n$ is counted by the double factorial $(2n-1)!! = (2n-1)\cdot (2n-3)\cdot\dots\cdot 3\cdot 1$, while on the other hand, any involution on a finite set of odd cardinality must fix at least one point.

Revised on August 20, 2015 05:59:57
by Noam Zeilberger
(176.189.43.179)