Given $E$ a vector space s.t. $\dim(E)=2$, and $G=\{g_1, g_2,... g_r\}$ a finite subgroup of $GL(E)$, $|G|=r$ (so $\forall g \in G, g^{r}=I$).
How to show that : $\mathrm{rk}(g_1, ..., g_r)=3 \implies \forall g \in G, g^2=I$
I showed that $\exists a, b$ s.t. $g^2 = a \cdot g + b\cdot I$ but I'm stucked.