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We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup $$\Gamma\subset G$$ , $$U\cap \Gamma$$ generates an abelian group. We show that for each n, there exists an integer $$w(n) > 0$$ , such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H 0, has an abelian subgroup of index $$\le w(n)$$ , where H 0 is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index $$\le w(n)$$ . We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed.

The object of this note is to interpret this relation in terms of group and module extensions. The basic idea is actually "classical" and goes back to Artin and Witt. Artin showed that one can associate to every group extension A~--,E--~ G, with A a k G-module, a module extension A~--,S~--* k 15 such that the cohomology class of the group extension splits in S~. Witt's contribution originates from a revealing discussion of the derivation u~-~u-1 from E to the group ring kE. We show, however, that the resulting module extension is equivalent to Artin's splitting module. Moreover, the assignment E~--~S~ induces the canonical correspondence between H 2 (G, A) and Ext~G(k 15, A). This simple observation translates, in a very convenient way, problems on group extensions to those on module extensions. In fact, Artin-Iyanaga [11] and Witt [14] implicitly made use of this in their famous work on the Principal Ideal Theorem. We give some further applications to illustrate the usefulness of this technique. We include a discussion of splitting theorems and of Frattini ("essential") extensions as well. Perhaps most striking is the module theoretic approach to Gaschiitz's splitting theorem. We then are concerned with the block theory of chief factors, resuming some known results from our point of view. Finally we lift splitting modules in the sense of modular representation theory, relating group extensions to semi-simple modules.