Differential and total cross sections for the quasifree reactions γp→ηp and γn→ηn have been determined at the MAMI-C electron accelerator by using a liquid deuterium target. Photons were produced via bremsstrahlung from the 1.5 GeV incident electron beam and energy tagged with the Glasgow photon tagger. Decay photons of the neutral decay modes η→2γ and η→3π0→6γ and coincident recoil nucleons were detected in a combined setup of the Crystal Ball and the TAPS calorimeters. The η-production cross sections were measured in coincidence with recoil protons, recoil neutrons, and in an inclusive mode without a condition on recoil nucleons, which allowed a check of the internal consistency of the data. The effects from nuclear Fermi motion were removed by a kinematic reconstruction of the final-state invariant mass and possible nuclear effects on the quasifree cross section were investigated by a comparison of free and quasifree-proton data. The results, which represent a significant improvement in statistical quality compared to previous measurements, agree with the known neutron-to-proton cross-section ratio in the peak of the S11(1535) resonance and confirm a peak in the neutron cross section, which is absent for the proton, at a center-of-mass energy W=(1670±5) MeV with an intrinsic width of Γ≈30 MeV.