TITLE: Do all representations of finite groups have one-dimensional subrepresentations?
QUESTION [12 upvotes]: 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 $<u>$ 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.

REPLY [13 votes]: It is possible that $u$ is zero.