Let $G$ be a group with $|G| = 8$.
By the Artin-Wedderburn Theorem, $\mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a decomposition?
Now, I know the dimensions on each side must be equal, so are the only possibilities $\mathbb{C}^{(8)}$, $M_2({\mathbb{C}}) \oplus M_2({\mathbb{C}}) $, $M_2({\mathbb{C}}) \oplus \mathbb{C}^{(4)}?$
I'm just not sure if anything else apart from $\mathbb{C}$ can appear as a division ring?
Say if we looked at $\mathbb{R}G$, then we could have things like $\mathbb{H} \oplus \mathbb{H}$, $M_{2}(\mathbb{C})$ and so on. Where only $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ itself can appear as division rings.