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Show that all irreducible representations of a finite group are finite dimensional. We know for any representation $V$ of a finite group $G$, there is a decomposition $V=V_1 \oplus V_2 \oplus.....\oplus V_n$ where $V_i$ s are all irreducibles. Now if we make $V$ irreducibles then just one $V=V_1$(say) will be there. But how can we show $V$ is finite dimensional?

Any help will be apreciated..

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Pick $v\in V$ nonzero and consider $\Bbb C[G]v$, the $\Bbb C$-span of $v$'s orbit under $G$.

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  • $\begingroup$ Ok then what I shall do?? $\endgroup$ – Ri-Li Sep 12 '15 at 17:53
  • $\begingroup$ @user152715 You should be done then. $\endgroup$ – Tobias Kildetoft Sep 12 '15 at 17:55
  • $\begingroup$ Ohh okay. This $\Bbb C Gv$ is a submodule of $V$ and it has only one. So it must be whole of $V$. Sorry... $\endgroup$ – Ri-Li Sep 12 '15 at 18:06
  • $\begingroup$ But where finiteness of $G$ is used here? $\endgroup$ – Ri-Li Sep 12 '15 at 18:08
  • $\begingroup$ @user152715: it means that $\mathbb{C}[G]$ is finite-dimensional. $\endgroup$ – Qiaochu Yuan Sep 12 '15 at 18:08

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