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I want to prove that $\sum_{g \in G} gv=0 $ on an irreducible representation $p: G \to GL(V)$ with $dim_{\mathbb{C}}(V) \geq 2$. In additional if i have an $h \in G$ and $gh=hg \ \ \ \forall g \in G $, is it possible to find that $p(h)= \zeta I $ for a root of unity $\zeta \in \mathbb{C}$ using the first part of my question?

Using Schur's Lemma i can find that for each $c \in Z(G)$ there exist a $z \in \mathbb{C} $ such that $p(c)v= zv $ but I'm completely lost.

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Suppose that $\sum_{g \in G} gv \neq 0$. Then the span of this vector is $G$-stable, not $0$, and not all of $V$ (since the dimension of $V$ is greater than $1$).

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