Relation between chief and compositions series of a group

Question

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

Answer 1

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.

Answer 2

Suppose that $G$ has a composition series. If you take any series (proper inclusions)

$$1 < N_1 < \cdots < N_t < N_{t+1} = G$$

with each $N_i$ normal in $G$, by Jordan-Hölder the length of this series is at most the composition length of $G$. Hence we can find a minimal normal subgroup $N \neq 1$ of $G$. Then we find a chief series of $G$ by considering $G/N$ and applying induction on the composition length of $G$.