Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 138 / 1 / a / Solution

Let $N={\sum }_{g\in G}g$ be the group norm element. This is a nonzero vector in the group algebra $kG$, and $kN$ is a trivial submodule of its left regular module. If $kG$ were a semisimple algebra, this submodule would have a $kG$-linear projection $\pi :kG\to kN$. Write $\pi (1)=cN$. Since $gN=N$ for every $g$, we would havewhereas a projection onto $kN$ satisfies $\pi (N)=N\ne 0$. Thus $\boxed{kG\text{ is not semisimple}}$.