Let $G$ be a finite group, $N$ be a minimal normal subgroup of $G$, and $M$ be a smallest normal subgroup of $G$. What's the difference between a smallest normal subgroup and a minimal normal subgroup? What's the definition of a smallest normal subgroup?
This definition needed for this theorem: Let $G$ be solvable with $\phi(G)=1$ and assume that each minimal normal subgroup has prime order or order $4$. Then every chief factor of $G$ has prime order or is $G$-isomorphic to a minimal normal subgroup of $G$ of order $4$.
Proof: Let $K/L$ be a chief factor of $G$. We proceed by induction on $|K|$. Let $M$ be the smallest normal subgroup of $K$ such that $K/M$ is nilpotent, and so $M \unlhd G$ and $M \leq L$.