For a positive integer g, let $$\mathrm {Sp}_{2g}(R)$$ Sp 2 g ( R ) denote the group of $$2g \times 2g$$ 2 g × 2 g symplectic matrices over a ring R. Assume $$g \ge 2$$ g ≥ 2 . For a prime number $$\ell $$ ℓ , we give a self-contained proof that any closed subgroup of $$\mathrm {Sp}_{2g}(\mathbb {Z}_\ell )$$ Sp 2 g ( Z ℓ ) which surjects onto $$\mathrm {Sp}_{2g}(\mathbb {Z}/\ell \mathbb {Z})$$ Sp 2 g ( Z / ℓ Z ) must in fact equal all of $$\mathrm {Sp}_{2g}(\mathbb {Z}_\ell )$$ Sp 2 g ( Z ℓ ) . The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.