I have recently begun learning about group algebras over finite fields but am still a little uncertain about these guys. So I was looking for some clarification and verification. Consider the Wedderburn decompositions for a group ring $\mathbb{F}_p[G]$ where $p$ prime and $G$ finite. Then we have three cases:
1) $p\nmid |G|$, then $\mathbb{F}_p[G]$ is semisimple by Maschke, and so $\mathbb{F}_p[G]\cong\prod_{i=1}^n M_{n_i}(D_i)$, where $D_i$ are all division algebras over $\mathbb{F}_p$ and hence isomorphic to $\mathbb{F}_p$;
2) $|G|=p^a$ for some $a\geq 1$, then $\mathbb{F}_p[G]$ is local with unique maximal ideal $I$= augmentation ideal. Therefore, $\mathbb{F}_p[G]/I\cong \mathbb{F}_p$;
3) $|G|=kp^a$ for some $a\geq 1$ and $k,p$ coprime. In this case, $\mathbb{F}_p[G]$ is a Frobenius algebra with nilpotent radical and so $\mathbb{F}_p[G]/rad\cong\prod_{i=1}^n M_{n_i}(D_i)$, where I believe the $D_i$ are once more division algebras over $\mathbb{F}_p$ and hence again isomorphic to $\mathbb{F}_p$.
Is this all correct? Am I missing anything? Is there anything more we can say about $\mathbb{F}_p[G]$?