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

Classification of Subgroups of Symplectic Groups Over Finite Fields Containing a Transvection