Let $G$, $H$, and $K$ be groups. Under what conditions does $G\times K\cong H\times K$ imply $G\cong H$?
This can easily be shown to be true when they are finitely generated abelian groups, and we can find counterexamples when $K$ is not finitely generated ($\{0\}\times\mathbb{R}\cong\mathbb{R}\times\mathbb{R}$ but $\{0\}\ncong\mathbb{R}$), but under what other circumstances is it true or false?
It is true whenever $G\times K$ has a composition series. (It does not need to be abelian.) It is also true whenever it has an $X$-composition series for operators $X$. (In particular, it is also true under the weaker condition that $G\times K$ has a chief series.)
This is a consequence of the Krull-Schmidt theorem for operator groups.