  Differential and total cross sections for the quasifree reactions $\gamma
p\rightarrow\eta p$ and $\gamma n\rightarrow\eta n$ have been determined at the
MAMI-C electron accelerator 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 $\eta\rightarrow 2\gamma$ and $\eta\rightarrow 3\pi^0 \rightarrow
6\gamma$ and coincident recoil nucleons were detected in a combined setup of
the Crystal Ball and the TAPS calorimeters. The $\eta$-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 $S_{11}(1535)$ resonance and confirm a peak in the
neutron cross section, which is absent for the proton, at a center-of-mass
energy $W = (1670\pm 5)$ MeV with an intrinsic width of $\Gamma\approx 30$ MeV.
