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IntroductionThe group \(GL_2(\mathbb{F}_q)\)consists of invertible \(2 \times 2\)matrices with entries in the finite field \(\mathbb{F}_q\), where \(q=p^n\) for some prime \(p\) (throughout we exclude the case when\(p=2\) because it can be quitedifficult). As a \(\mathbb{F}_p\)-vector space, \(\mathbb{F}_q\) has dimension \(n\). The Galois group \(G(\mathbb{F}_q/\mathbb{F}_p)\) is cyclicand is generated by the Frobenius map.The field \(\mathbb{F}_q\) itself isalready pretty complicated, let alon...