Finite matrix group

Let $\mathcal{G}$ be a finite subgroup of ${\rm GL}_n (\mathbb{C})$ (this is the group of the $n \times n$ invertible matrices over $\mathbb{C}$). If $\sum \limits_{g \in \mathcal{G}} {\rm Tr}(g)=0$ then prove that $\sum \limits_{g \in \mathcal{G}} g =0$.

Solution

Let us suppose that $|\mathcal{G}| = \kappa$ and $x= \frac{1}{\kappa} \sum \limits_{g \in \mathcal{G}} g$. We note that for every $h \in \mathcal{G}$ the depiction $\varphi: \mathcal{G} \rightarrow \mathcal{G}$ such that $\varphi(g)=h g$ is $1-1$ and onto. Thus:

\begin{align*}
x^2 &=\left ( \frac{1}{\kappa} \sum_{g \in \mathcal{G}} g \right )^2 \\
 &= \frac{1}{\kappa^2} \sum_{g \in \mathcal{G}} \sum_{h \in \mathcal{G}} gh\\
 &= \frac{1}{\kappa^2} \sum_{g \in \mathcal{G}} \sum_{h \in \mathcal{G}} g\\
 &= \frac{1}{\kappa} \sum_{h \in \mathcal{G}} \left (\frac{1}{\kappa} \sum_{g \in \mathcal{G}} g \right ) \\
 &= \frac{1}{\kappa} \sum_{h \in \mathcal{G}} x\\
 &= \frac{1}{\kappa} \kappa x \\
 &=x
\end{align*}

Thus the matrix $x$ is idempotent , thus its trace equals to its class. (since we are over $\mathbb{C}$ which is a field of zero characteristic.) Hence

$${\rm rank} \;(x) = {\rm trace} \;(x) = \frac{1}{\kappa} \sum_{g \in \mathcal{G}} {\rm trace} \; (g) =0$$

This implies that $x=0$ hence $\sum \limits_{g \in \mathcal{G}} g =0$.

The exercise can also be found at mathematica.gr