Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 19F / b / iii / 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 $\eta (z)\ne 1$: 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 no such tensor-generating irreducible $W$ exists.