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IMC 2021 P6: No injective homomorphism from GL_2(Z/pZ) to S_p

Source: IMC 2021 P6

August 5, 2021
IMC 2021group theorynumber theoryprime numbersabstract algebra

Problem Statement

For a prime number pp, let GL2(Z/pZ)GL_2(\mathbb{Z}/p\mathbb{Z}) be the group of invertible 2×22 \times 2 matrices of residues modulo pp, and let SpS_p be the symmetric group (the group of all permutations) on pp elements. Show that there is no injective group homomorphism ϕ:GL2(Z/pZ)→Sp\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p.
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