Let $S$ be a finite set and $f$ a map with $f (f x) = x$ and $f x ≠ x$ for every $x ∈ S$, and which sends $S$ into itself. Then the elements of $S$ come in pairs ${x, f x}$, so $S$ has an even number of elements. This is the elementary counting principle behind parity arguments of the kind "the number of representations is even because negating one coordinate is a fixed-point-free involution".