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Schur's Theorem and its generalisation, Baer's Theorem, are distinguishedresults in group theory, connecting the upper central quotients with the lowercentral series. The aim of this paper is to generalise these results in twodifferent directions, using novel methods related with the non-abelian tensorproduct. In particular, we prove a version of Schur-Baer Theorem for finitelygenerated groups. Then, we apply these newly obtained results to describe the$k$-nilpotent multiplier, for $k\geq 2$, and other invariants of groups.

Some generalisations of Schur's and Baer's theorem and their connection with homological algebra