# Symmetric group question

## Homework Statement

$sgn(\sigma)=0\iff\sigma$ is not a bijection.

## The Attempt at a Solution

$(\rightarrow)$Let $sgn(\sigma)=0$. Then, $\Pi_{1\leq i<j\leq n}\frac{\sigma(j)-\sigma(i)}{j-i}=0.$. For some $i$ and $j$, $i\neq j$, $\sigma(i)=\sigma(j)$. Thus, $\sigma$ is not an injection.

$(\leftarrow)$ $\sigma$ not a bijection $\rightarrow$ it is not an injection, same argument as above, $sgn(\sigma)=0$.

I think I went wrong somewhere. Any ideas?

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