# 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