# Show that all irreducible representations of a finite group are finite dimensional.

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..

## 1 Answer

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

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