I was having a conversation today regarding Maschke's theorem that says that the complex algebra $\mathbb{C}G$ is semisimple. Moreover if the group is abelian then every simple $\mathbb{C}G$-module is of dimension 1 and $\mathbb{C}G$ itself is a product of $|G|$ copies of $\mathbb{C}$.
My question is; are there any conditions that we can impose on $G$ in order for the group algebra to be simple? Obviously $G$ must be nonabelian and its order must be a square but apart from that?
On
$\mathbb{C}[G]$ is simple if and only if there is only one irreducible representation up to isomorphism. But there is always the trivial representation, so it must be the only representation. Since this is a $1$-dimensional representation, it appears with multiplicity $1$ in $\mathbb{C}[G]$, hence $|G|=1$.
The group algebra $\Bbb CG$ is simple if and only if $|G|=1$.
For nontrivial $G$ the kernel of the map $\phi:\Bbb CG\to\Bbb C$ taking each $g\in G$ to $1$ is a nontrivial ideal of $\Bbb CG$ (its augmentation ideal). So $\Bbb CG$ is not simple.