IMC 2021 · Problem 6

For a prime number $p$, let $\mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\varphi \colon \mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z}) \to S_p$.