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I've been stuck on this homework question for a couple of days. Any hints would be much appreciated.

Let $G$ be a finite group and $\rho : G \to \mathrm{GL}(V)$ be a faithful representation, with $V$ a vector space over the field $F$. Suppose $Z(G) = \{ e \}$, and $H \le G$ is a subgroup with $Z(H) \ne \{ e \}$. Then, I need to show that the restriction $\rho|_H : H \to \mathrm{GL}(V)$ is reducible.

I have no idea how to proceed, but apparently Schur's lemma is relevant. I'm not sure if I'm missing some conditions, like say $F = \mathbb{C}$. I can see, for instance, that the image of $Z(H)$ under $\rho$ is a family of commuting automorphisms, and so can be simultaneously diagonalised. If I assume that $\rho|_H$ is irreducible then Schur's lemma implies that in fact $\rho(z) = \lambda_z \mathrm{id}$ for some $\lambda_z \in \mathbb{C}^\times$ for each $z \in Z(H)$. This should lead to a contradiction somewhere, but I'm not seeing it... I also notice I haven't used the fact that the parent group $G$ is centreless.

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Yes, Schur's lemma is relevant. If rho is irreducible, then Z(H) acts by scalars. But scalars are in the center of GL(V)... – Qiaochu Yuan Jan 8 '11 at 13:23
Dur. I feel stupid now. $\rho$ is faithful so there isn't anything in the the centre of $\mathrm{GL}(V)$ in the first place. Thanks! – Zhen Lin Jan 8 '11 at 13:31

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