Two types of remainders of topological groups

summary:We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel'skii, {\it Some connections between properties of topological groups and of their remainders\/}, Moscow Univ. Math. Bull. 54:3 (1999), 1--6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions