What are some examples of non-identity bijections $f: X \to X$ such that $f^{-1} = f$

Question

One example I can think of is $f: \mathbb{Z_2} \to \mathbb{Z_2}$ given by $f(1) = 0$ and $f(0) = 1$.

Answer 1

In a sense, every such bijection is going to look the same. If $a,b \in X$, then we will either have things that look like $f(a) = a$ or $f(a) = b, f(b) = a$ (which I'm going to refer to as a single 'transposition.'

But this gives us infinitely many examples to choose from, even just in $\mathbb{Z}$. You might let $f$ be the identity on every element except, say, $1$ and $5$, such that $f(1) = 5, f(5) = 1$. Or you can have as many transpositions as you'd like.

Answer 2

Interesting example. $f(x)=-x, x\in \mathbb{R}$, or $f(x)=\frac{1}{x}, x\in \mathbb{R}\backslash \{0\}$ would come to mind first for me. In general such a function can clearly be defined on any group-like structure with inverses, just by defining a function that takes every element to its inverse, and the identity (if it exists) to itself.