Composite of Permutations is Permutation

Theorem

Let $f, g$ are permutations of a set $S$.

Then their composite $g \circ f$ is also a permutation of $S$.

Proof

This follows from the fact that a permutation is a bijection.

The domain and codomain are coincident.

The result follows from Composite of Bijections is Bijection.

$\blacksquare$