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=V1V2.....Vn where Vi s are all irreducibles. Now if we make V irreducibles then just one V=V1(say) will be there. But how can we show V is finite dimensional?

Any help will be apreciated..

Answer

Pick vV nonzero and consider C[G]v, the C-span of v‘s orbit under G.

Attribution
Source : Link , Question Author : Ri-Li , Answer Author : whacka

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