nLab
involution

Contents

Idea

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).

Properties

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

fg=f(fgfg)g=(ff)gf(gg)=gff g = f (f g f g ) g = (f f) g f (g g) = g f
(fg)(fg)=f(gf)g=f(fg)g=(ff)(gg)=1(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:XXf : X \to X on a finite set of even cardinality |X|=2n|X| = 2n is counted by the double factorial (2n1)!!=(2n1)(2n3)31(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)