How does one compute a group modulo a torsion group?

Question

Let's say I have some group $G$ and a subgroup $H$ such that $H$ is a torsion group (i.e. $\forall h \in H$, $h$ has finite order. How do I compute the factor group $\frac{G}{H}$? What effect does the order of each torsion element have on it?

Answer 1

Since your groups are most likely abelian and finitely generated, you can factor $G$ into the direct sum of its torsion group and a free group. This reduces any such problem to a problem about finite abelian groups.