= Fitting subgroup centralizes the socle
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For a <finite group>, $F(G)$ centralizes its <group socle>. A nonabelian <minimal normal subgroup> meets $F(G)$ trivially: otherwise it is nilpotent, and its nontrivial characteristic center makes it abelian. Their <group commutators> therefore vanish. An abelian <minimal normal subgroup> $M$ is an <elementary abelian p-group>; $M\le O_p(G)$ and $M\cap Z(O_p(G))\ne1$ by the fixed-point counting of a <finite p-group> on $M$. Minimal normality makes $M\le Z(O_p(G))$, and other <p-cores> commute with $M$.
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