Find all groups $G$ such that the corresponding algebra $\mathbb{C}[G]$ is central
I know that since $\mathbb{C}$ is algebraically closed we have: $\mathbb{C}[G]=\prod_{i=1}^sM_{n_s}(\mathbb{C})$ so in order for $\mathbb{C}[G]$ to be central we must have $\mathbb{C}[G]=M_n(\mathbb{C})$. This implies $|G|=n^2$ and $G$ non abelian or $n=1$ which gives us the trivial group. However if $n>1$ we would have a non trivial group with exactly 1 conjugacy class so the group in question is exactly the trivial one. Have I gotten it wrong somewhere?
Seems ok, but if you're sure that it has to be a single matrix ring, it's faster to just note that $M_n(\mathbb C)$ is simple and $\mathbb C[G]$ is only simple when its augmentation ideal is $\{0\}$, meaning $G=\{1\}$.