Classification of subgroups of symplectic groups over finite fields containing a transvection

In this note, we give a self-contained proof of the following classification (up to conjugation) of finite subgroups of GSpn(Fl) containing a nontrivial transvection for l ≥ 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Spn(Fl). This result is for instance useful for proving ‘big image’ results for symplectic Galois representations.

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Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.