International Mathematics Competition
for University Students
2021

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

Hint: First find what the monomorphism must do with elements of order \(\displaystyle p\).

  Solution