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Let $G$ be a group, $E$ be a vector space over field $K$ and $\rho : G \rightarrow \operatorname{GL}(E)$ a semisimple $K$-representation of $G$. Let $H \lhd G$ be a finite-index normal subgroup of $G$.

How do I show that $\operatorname{Res}^G_H(\rho)$, a restriction of $G$ to $H$ is also semisimple as a representation of $H$?

Sigur
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user113988
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  • I don't think this is true in general. Take $\mathbb{Z}_3 \triangleleft S_3$ and restrict the 2-dimensional irreducible representation of $S_3$. It will be the direct sum of two 1-dimensional representations since $\mathbb{Z}_3$ is abelian. – Turion Dec 09 '14 at 16:08
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    @Turion that is semisimple. – Kenta S Jan 01 '23 at 04:36
  • True, no idea what I was thinking those 8 years ago. – Turion Jan 02 '23 at 08:12

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