Group G Preserving a Flag is Solvable
Solution 1:
You could do it by induction on $k$. It is true by assumption for $k=1$.
For $k>1$, the image of $\rho_k$ is solvable by assumption.
The kernel $K$ of $\rho_k$ acts on the flag $V_0 \subset V_1 \subset \cdots \subset V_{k-1}$ of length $k-1$ and the image of $\phi_i$ restricted to $K$ is solvable for $1 \le i \le k-1$, so $K$ is solvable by inductive hypothesis.
Now $G/K \cong {\rm Im}(\phi_k)$ with $K$ and ${\rm Im}(\phi_k)$ solvable, so $G$ is solvable.