# International Mathematics Competition for University Students 2021

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IMC 2021
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## IMC2021: Day 2, Problem 6

Problem 6. For a prime number $\displaystyle p$, let $\displaystyle \mathrm{GL}_2(\ZZ/p\ZZ)$ be the group of invertible $\displaystyle 2 \times 2$ matrices of residues modulo $\displaystyle p$, and let $\displaystyle S_p$ be the symmetric group (the group of all permutations) on $\displaystyle p$ elements. Show that there is no injective group homomorphism $\displaystyle \varphi : \mathrm{GL}_2(\ZZ/p\ZZ) \to S_p$.

Thiago Landim, Sorbonne University, Paris

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