A theorem on remainders of topological groups

It has been established in [7-9] that a non-locally compact topological group G with a first-countable remainder can fail to be metrizable. On the other hand, it was shown in [6] that if some remainder of a topological group G is perfect, then this remainder is first-countable. We improve considerably this result below: it is proved that in the main case, when G is not locally compact, the space G is separable and metrizable. Some corollaries of this theorem are given, and an example is presented showing that the theorem is sharp.