Exercise 4, § 1

Let $G$ be a commutative group, written additively, and let $p$ be a prime number. Denote by $G \left[ \frac{1}{p} \right]$ the commutative group $G \otimes_{\mathbf{Z}} \mathbf{Z} \left[ \frac{1}{p} \right]$. Show that if $K$ is a field of characteristic $p$, then the perfect closure of the group algebra $K[G]$ is the group algebra $K^{p^{-\infty}} \left[ G \left[ \frac{1}{p} \right] \right]$.