Every finite group has a chief series A chief series in a group $G$ is a series of normal subgroups such that
$1=N_0 \triangleleft N_1 \triangleleft ... \triangleleft N_n=G$,
for which each factor $N_{i+1} / N_i$ is a minimal (non-trivial) normal subgroup of $G/N_i$.
I am trying to prove that every finite group has at least one chief series and my idea is to do it by induction, but I have not managed to make it work yet.
 A: We'll prove by induction on size of $G$. If $|G|$ is 1 or 2, then $\{0\} \triangleleft G$ is a chief serie.  Let $G$ be a group of size $n$ and let $N$ be a minimal normal subgroup of $G$, if $N$ is $\{0\}$, then $N \triangleleft G$ is a chief series. So suppose $N$ is not $\{0\}$, then by induction, $G/N$ has a chief series $1=N_0 \triangleleft N_1 \triangleleft N_2...\triangleleft N_k=G/N$.  Let $\phi: G \rightarrow G/N$ be the canonical quotient homomorphism. Then we claim that $1 \triangleleft \phi^{-1}(1) \triangleleft \phi^{-1}(N_1)\triangleleft ...\triangleleft \phi^{-1}(G/N)=G$ is a chief serie for $G$. One can check $\phi^{-1}(N_i)\triangleleft \phi^{-1}(N_{i+1})$ because it is a property of surjective homomorphism. Also, as $N_{i+1}/{N_i}$ is minimal no trivial normal subgroup of $\frac{G/N}{N_{i}}$, $\phi^{-1}(N_{i+1}/N_i)$ is the smallest normal subgroup of  $\phi^{-1}(G/N_{i})$. And as $N$ is minimal normal in $G$, it is minimal normal in $\phi^{-1}(N_1)$. I hope this is correct.
