Relation between chief and compositions series of a group Is there an example of a group with a composition series (of finite length) but without a chief series (of finite length)? Is there an example of a group with a chief series (of finite length) but without a composition series (of finite length)? Definition of composition series Definition of chief series

Let $G$ be the semidirect product ${\mathbb Q}^2 \rtimes {\rm SL}(2,{\mathbb Q})$ with the natural action. Then $G$ has a chief series $1 < {\mathbb Q}^2 < {\mathbb Q}^2 \rtimes \\{\pm I_2\\} < G$ and, since the action on ${\mathbb Q}^2$ is irreducible and ${\rm PSL}(2,{\mathbb Q})$ is simple, these are the only normal subgroups of $G$. Since ${\mathbb Q}^2$ has no finite composition series, neither does $G$.

It would guess that having a finite composition series implies having a finite chief series.