Given a representation $V$ over $\mathbb{C}$ of dimension $n$ for a finite group $G$, suppose for all $g\in G$, we have $\dim(V_g)\geq n/2$, where $V_g$ is the subspace of $V$ that consists of elements that are fixed by the action of $g$. Prove that there exists a non-zero element $v\in V$ that is fixed by the entire group.
I can prove using some simple counting arguments that for any two elements of the group, we can find a non-zero element that is fixed by both the elements. But, I don’t know how to extend that, or how that may lead to a proof?
If $\text{dim}(V_g) \ge n/2$ then the corresponding character value $\chi_g$ satisfies $\text{Re}(\chi_g) \ge 0$. This is since $g$ acts with eigenvalue 1 on the fixed subspace and all other eigenvalues are roots of unity.
In particular the assumption tells us that $\text{Re}(\chi_g) \ge 0$ for all $g$ and its always strictly positive when $g =e$. Hence $\text{Re}(\langle\chi, \mathbf1 \rangle) = \text{Re}(\frac{1}{|G|}\sum_{g \in G} \chi_g )> 0$. But this is an integer so the whole thing is positive. So $V$ contains a copy of the trivial representation.