If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring.

Question

If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring.

My failed attempt: Since $G$ is finite and $p$ divisdes $|G|$ then $G$ has an element of order $p$. I wanted to use the cyclic subgroup generated by this element to create an ideal of $k[G]$ and show that it could not be produced by an idempotent element but I got stuck.

Answer 1

Hint:

Square $\sum_{g\in G}g$, and observe that it is central.