TITLE: The Fitting subgroup centralizes minimal normal subgroups in finite groups
QUESTION [7 upvotes]: Let $G$ be a finite group:
If $N$ is a minimal normal subgroup of $G$, then $F(G) \leq C_G(N)$.
Here $C_G(N)$ denotes the centralizer of $N$ in $G$, and
$F(G)$ denotes the Fitting subgroup of $G$.

REPLY [5 votes]: If $N$ is abelian, then $N \leq F$ and $[F,N] < N$ is still normal, so $[F,N]=1$ as claimed. If $N$ is non-abelian, then $[F,N] \leq F \cap N = 1$ as claimed.
You may try to prove sort of a converse: $F(G)$ is the intersection of the centralizers, not just of the minimal normal subgroups, but of all $K/L$ where $K$ is a minimal normal subgroup of $G/L$.