Group ring C[Z/n] and Artin-Wedderburn decomposition

Question

I am trying to answer the following questions, which I assume follow on from eachother each other;

1. Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings.
2. For abelian groups $G_1$, $G_2$, $\mathbb C$[$G_1$] $\cong$ $\mathbb C$[$G_2$] iff |$G_1$| = |$G_2$|.
3. Write $\mathbb C$[$D_8$] as a product of simple $\mathbb C$-algebras, and show that $\mathbb C$[$D_8$] $\cong$ $\mathbb C$[$Q_8$] (Dihedral and Quaternion groups).

For 1, at the moment I only have decompositions for n = 2 and 3 (from here), but I do not know how to generalise it for arbitrary n. And as one might imagine, I really don't know how to tackle the next 2 questions.

Answer 1

Maschke's theorem says that $\Bbb C[\Bbb Z/(n)]$ is a semisimple ring.

Since the cyclic group is abelian, we are talking about a commutative $\Bbb C$-algebra, and as a corollary of the Artin-Wedderburn theorem, a commutative semisimple $\Bbb C$-algebra is a product of finite field extensions of $\Bbb C$.

Since $\Bbb C$ is algebraically closed, there are no nontrivial finite field extensions of $\Bbb C$, so that means the group algebra is simply a finite product of copies of $\Bbb C$.

So to generalize:

If $G$ is finite and abelian, then $\Bbb C[G]$ is isomorphic to the product ring $\Bbb C^n$ where $|G|=n$.

This should be enough for you to solve 1 and 2.

For 3, $D_8$ is a nonabelian group, so we are in a new situation.

You should still begin the same way by analyzing the group algebra as a semisimple ring. The matrix rings in the decomposition will all be over $\Bbb C$, for exactly the reasons mentioned before. Since it's noncommutative, it must contain at least one nontrivial matrix ring. By counting dimensions and looking for idempotents, determine what the matrix rings must be.

Here's another useful thing someone taught me along the way about idempotent hunting in group rings: if the field has characteristic zero and $H<G$, then $\frac{1}{|H|}\sum_{h\in H}h$ is idempotent in $F[G]$. If $H$ is normal in $G$, then the idempotent is central (meaning it determines a splitting of the ring into two smaller rings.)

$D_8$ has a normal subgroup of size $4$ which is cyclic... so that half of the ring we have analyzed. The other half must be noncommutative. How do you fit a noncommutative $\Bbb C$ algebra into $4$ dimensions?

Similarly, you can look for abelian subgroups of $Q_8$, and luckily all subgroups of $Q_8$ are normal.