IMC 2021 · Day 2 · P6

Hint: First find what the monomorphism must do with elements of order $p$.

For $p = 2$, just note that $\operatorname{GL}_2(\mathbb{Z}/2\mathbb{Z})$ has more than $2 = |S_2|$ elements.

From now on, let $p$ be an odd prime and suppose that there exists such a homomorphism.

The matrix $$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$ has order $p$ and commutes with the matrix $$B = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$$ of order 2, hence $AB$ has order $2p$. But there is no permutation in $S_p$ of order $2p$ since only $p$-cycles have order divisible by $p$, and their order is exactly $p$.