 Central commutator has no torsion from an abelian Sylow subgroup

If a Sylow p-subgroup $P$ is abelian, a hypothetical central subgroup $Q\le G^{\prime }$ 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 $G^{\prime }$. This contradiction proves the displayed restriction.