The very efficient end-bridging Monte Carlo (EBMC) method has been employed in order to simulate an amorphous, polydisperse 80-chain large C-156 polyethylene (PE) system in atomistic detail over a wide range of temperatures (from 600 down to 150 K) and determine its glass transition temperature (T-g). Two sets of simulations have been performed: one with a bulk, isotropic sample and the,other with a thin film in which all the 80 PE chains were grafted on a hard substrate on one side (corresponding to a high grafting density equal to or = 1.75 nm(-2)) and exposed to vacuum on the other side. In the simulations, a united-atom model was employed for PE ensuring that only the purely amorphous phase of PE was simulated at all temperatures. In all cases, very long simulations were carried out in order to give enough time for the system to relax at all length scales. For all temperatures studied, the longest relaxation time was found to be present by descriptors associated with the system's long-range conformational characteristics. In contrast, more local, internal structural features were always faster in equilibrating. As a result, the time autocorrelation function for the chain end-to-end unit vector, f(u)(t), was found to drop to zero and then clearly fluctuate around this value only for temperatures higher than about 220 K for both systems. For lower temperatures, f(u)(t) did not relax completely, even after 2 x 10(7) CPU seconds. Additional volumetric simulation data demonstrated a sharp change in the density and potential energy of both systems in the neighborhood of the 230 K, which are considered as features of the glass transition for amorphous PE. The T, value suggested by the present EBMC simulations for amorphous (bulk or grafted) PE is (230 +/- 10) K, which is consistent with the value of 237 K measured experimentally by Wunderlich [J. of Chem. Phys. 1962, 37, 1203] and Loufakis and Wunderlich [J. Phys. Chem. 1988, 92, 4205] for PE in the limit of zero crystallinity. Further, the predicted change in the heat capacity at constant pressure at the glass transition is Delta c(p) = 1.2 x 10(-4) kcal g(-1) K-1, which is very close to the value of 1.5 x 10(-4) kcal g(-1) K-1 measured experimentally by Wunderlich [J of Chem. Phys. 1962, 37, 1203]. Additional results on the temperature dependence of the conformational and structural properties in the two PE systems are also reported and discussed in detail.