I am reading Alperin's book "Local Represenation Theory", and I am stuck on exercise 4 of page 28.
Throughout the previous exercises from this section one proves the following:
Let $G=C_{p}\times C_{p}$ be a group which is the direct product of two cyclic groups of order $p$ and let $k$ be a field with char$(k)\mid p$. Also, denote $x , y$ as generators of each coordinate of $G$.
Now, let $W$ be a two-dimensional vector space with basis $\{u,v\}$, then for any given $\alpha, \beta\in k$ there exists a unique $kG-$module structure on $W$ such that: $$xu=u+\alpha v ; xv=v $$ $$yu=u+\beta v ; yv=v$$
We denote this $kG-$module by $V_{\alpha, \beta}$.
Furthermore, for some $\gamma, \delta \in k$ we get that $V_{\alpha, \beta}$ is isomorphic to $V_{\gamma, \delta}$ if and only if $(\alpha, \beta)$ and $(\gamma, \beta)$ are proportional.
I have managed to solve everything up to this point, it is the next question that I am struggling with. It says:
Establish that any two-dimensional $kG-$module is isomorphic with some $V_{\alpha, \beta}$.
Thanks a lot in advance.