I heard recently that the $p$-rank of a finite abelian group (the number of cyclic components of size $p^n$) is given by $\dim_{\mathbb{F}_p}(\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G])$, which is a result I hadn't seen before. I'm struggling to see the connection.
So far I have shown that $\mathbb{F}_p \otimes_\mathbb{Z} \mathbb{Z}[G] \cong \mathbb{F}_p[G]$. How can I find the dimension of $\mathbb{F}_p[G]$, and relate that to the $p$-rank of $G$?