Involution

Remarks

Example: complex conjugation $z \mapsto \bar{z}$ on $\C$ is an involution, because $\overline{\overline{a + bi}} = \overline{a - bi} = a + bi$.

Non-example: the map $x \mapsto x + 1$ on $\Z$ is not an involution, because applying it twice gives $x + 2 \ne x$.

Every involution is a bijection, and it is its own inverse. The identity map is an involution. On a finite set, the number of points not fixed by an involution is even, since they come in pairs $\set{x, f(x)}$. In a group, the map $x \mapsto x^{-1}$ is an involution, because $(x^{-1})^{-1} = x$.