Let $\rho\colon G\longrightarrow GL_{3}(\mathbb{C})$ be a representation of a finite group. Show that $\rho$ is irreducible if and only if there is no common eigenvector for the matrices $\rho_{g}$ with $g \in G$.
My question is how can I do the demostration and what if I aplied this to dim $2$, I'm starting to study the subject of representation of groups and I'm using the book by Benjamin Steinberg: Representation Theory of Finite Groups.
If $v$ is a common eigenvector for the matrices $\rho_g$ ($g\in G$), then clearly $\mathbb C v$ is an invariant subspace of $\mathbb C^3$ and therefore $\rho$ is not irreducible.
On the other hand, if $\rho$ is not irreducible, you can write $\mathbb C^3$ as the direct $V\oplus W$ where $V$ and $W$ are non-trivial subspaces of $\mathbb C^3$ and $V$ and $W$ are stable with respect to the action $\rho$. But, since $\dim\mathbb C^3=3$, one of the spaces $V$ and $W$ must be $1$-dimensional. Assume that it is $V$. Then $V=\mathbb C v$ for some non-null vector $v$, which will be then a common eigenvector for the matrices $\rho_g$.
Now, do you think that this argument also works in $\mathbb C^2$ instead of $\mathbb C^3$?