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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.

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up vote 10 down vote accepted

It is possible that $u$ is zero.

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Indeed. Thank you! –  Artem Sep 3 '12 at 17:05
@Artem, You could have found this out by yourself by picking any irreducible representation of dimesion larger than one of any group! You should always look at examples. –  Mariano Suárez-Alvarez Sep 3 '12 at 17:29

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