= Solution
Let $\chi$ be the central <one-dimensional character> of a proposed $W$. Since $Z$ is not cyclic, part (a) makes this <one-dimensional character> nonfaithful. Choose a nonidentity $z\in\ker\chi$. It acts as identity on every <tensor power> $W^{\otimes n}$ and therefore on every summand. There is a <one-dimensional character> $\eta$ of $Z$ with $\eta(z)\ne1$: decompose the finite <abelian group> into cyclic factors and choose a primitive <one-dimensional character> on a factor where $z$ has a nonidentity component. By part (b)(ii), some <irreducible representation> $V$ of $G$ has central <one-dimensional character> $\eta$ on $Z$. This $V$ cannot occur in any <tensor power> of $W$, because $z$ acts nontrivially on it. Therefore \b[no such tensor-generating irreducible $W$ exists].
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