Let V be a representation of a finite group G, and $v\in V$ - a nonzero vector. Put $u = \sum_{g\in G} gv.$ Then for any $g\in G$ we have $gu = u$ and therefore $$ is a subrepresentation of V.
I know there is an error here since there are irreducible representations of finite groups which are not one dimensional, but I can't see it. Could someone point it out?
Thank you.