  Based on a data sample of (1.0087+-0.0044)x10^10 Jpsi events collected by the
BESIII detector at the BEPCII accelerator, the absolute branching fraction (BF)
of the decay Jpsi->gamma eta is measured with high precision using events in
which the radiative photon converts to e+e-. Using the measured absolute BF of
Jpsi->gamma eta, the absolute BFs of four dominant eta decay modes are measured
for the first time. The results are B(Jpsi->gamma eta) =
(1.067+-0.005+-0.023)x10^-3, B(eta->gamma gamma) = (39.86+-0.04+-0.99)%,
B(eta->pi0pi0pi0) = (31.96+-0.07+-0.84)%, B(eta->pi+pi-pi0) =
(23.04+-0.03+-0.54)%, and B(eta->pi+pi-gamma) = (4.38+-0.02+-0.10)%, where the
first and second uncertainties are statistical and systematic, respectively.
The results are consistent with the world average values within two standard
deviations.
