Existence of nontrivial normal subgroups in solvable finite groups Let $G$ be a finite solvable group. Is it true that $G$ has a minimal nontrivial normal subgroup, i.e. a subgroup $N$ with $N\trianglelefteq G$, $N\neq 1$ and with the property that if $K\trianglelefteq G$ and $K\le N$ then either $K=1$ or $K=N$. I've tried inducting on $|G|$ and on the solvable lenght of $G$ but I got nothing.

Start with $N_{0}=G$. Once you have $N_{i}$, look to see if there are any nontrivial proper subgroup of $N_{i}$ that is also a normal subgroup of $G$. If there is, then let that be $N_{i+1}$ and repeat. If not, then you are done, and $N_{i}$ is what you get. The process must halt, as $|G|$ is finite.

Really, I do not see how being solvable is relevant at all.