= Central commutator has no torsion from an abelian Sylow subgroup
{title2=$p\nmid|G'\cap Z(G)|$}
If a <Sylow p-subgroup> $P$ is abelian, a hypothetical central subgroup $Q\leq G'$ of order $p$ has a nontrivial linear character extending to $P$. Inducing this extension to $G$ gives a character of degree prime to $p$, hence an irreducible constituent of such degree. On $Q$ its <determinant character> is a nontrivial power of the chosen linear character, but every determinant character is trivial on $G'$. This contradiction proves the displayed restriction.
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