位相幾何学とグループ理論

最近整数ホモロジーを学びながらグループ理論の話にぶつけました。
Basically, a Torsion subgroup is a subgroup of elements x such that there exists a nonzero n such that nx=0. I find such a group difficult to imagine, with the exception of idemgroups. It seems that the integers are torsion-free (as only 0 would qualify). The theorem which seems most relevant is that any finitely generated group G can be written as the direct sum of its torsion subgroup T and L^r, where L^r is the set of all r-tuples of integers, it seems. I am not sure, though, exactly how this theorem relates to integral homology.