Relations among the invariants of topological groups and their subspaces

In this paper we study topological properties of topological groups and, first of all, cardinal invariants of topological groups. Many of the relevant questions are subsumed under the following general scheme: how does the compatibility of the topology with the group structure reflect on the relations among the invariants of this topology?
We use the notation and terminology of $ \lbrack 4\rbrack$. Cardinal invariants of a topological group are understood to mean those of its underlying space, which is assumed throughout to be completely regular and $ T_1$. Proofs are given in condensed form or omitted altogether.