= Minimal normal subgroups of finite solvable groups are elementary abelian
A nontrivial <minimal normal subgroup> $N$ of a finite <solvable group> is abelian: its <commutator subgroup> is characteristic, and minimality plus solvability excludes $N'=N$. A nontrivial <Sylow subgroup> of this <abelian group> is characteristic, so minimality makes $N$ a p-group. The <characteristic subgroup> of pth powers is proper, hence trivial by minimality. Therefore $N$ is an <elementary abelian group>.
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