= Solution
Let $w=\sum_{g\in G}g$. This is nonzero since its <basis> coefficients are all one, and $gw=w$ for every $g\in G$. Hence $kw$ is a one-dimensional left ideal of $kG$.
If the regular <module> is semisimple, $kw$ is a direct summand. Let $\pi:kG\to kw$ be the corresponding $kG$-linear projection, so $\pi(w)=w$. Write $\pi(1)=cw$ for $c\in k$. Then
$$
w=\pi(w)=\sum_{g\in G}\pi(g)=\sum_{g\in G}g\pi(1)=c|G|w.
$$
Because $w\ne0$, $c|G|=1$ in $k$. Thus
$$
\boxed{|G|\ne0\text{ in }k.}
$$
Together with the <Maschke theorem>, this gives the exact <semisimple algebra> criterion for a <group algebra>.
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