A group ring has finite length

Question

I have been given a version of Maschke's Theorem to prove:

Let $k$ be a field and $G$ a finite group s.t. $|G|$ is non-zero in $k$. Show that the group ring $k[G]$ is a semi-simple ring.

A hint to start proving this is to "show $k[G]$ has finite length" (there's some more hints from here using the composition series of the group ring). I'm not sure how to go about showing $k[G]$ has finite length; what sort of chain of submodules can I construct given how general this is? I really just can't see how to start.

Any help would be appreciated!

Answer 1

Here is a way of going about it without the hint:

Let $V$ be a finite dimensional $\mathbb{k}G$-module and $W \leq V$. Choose an idempotent $e \in \text{End}_\mathbb{k}(V)$ such that $eV=W$. Let $f=\frac{1}{|G|}\sum_{g \in G} geg^{-1}$. Then if $h \in G$, what is $hf$?

$$hf=\frac{1}{|G|} \sum_{g \in G} h(geg^{-1})=\frac{1}{|G|} \sum_{g \in G} (hg)e(g^{-1}h^{-1})h=\frac{1}{|G|}\sum_{g \in G} (hg)e(hg)^{-1}h=fh$$

Then what can we say about $f$?

$f \in \text{End}_{\mathbb{k}G}(V)$

Now as $W \leq V$, we know what about $fV$ and $f|_W$?

$fV \leq W$ and $f|_W=1_W$

Then $f$ is idempotent with...

$fV=W$

so that $V=W \oplus (1-f)V$