Injection from Finite Set to Itself is Surjection/Corollary
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Theorem
Let $S$ be a finite set.
Let $f: S \to S$ be an injection.
Then $f$ is a permutation.
Proof
From Injection from Finite Set to Itself is Surjection, $f$ is a surjection.
As $f$ is thus both an injection and a surjection, $f$ is a bijection by definition.
Thus as $f$ is a bijection to itself, it is by definition a permutation.
$\blacksquare$
Sources
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts