21.60 (2026)

Let $G$ be a finite group, $\mathbb{Z}_{(p)}$ the localization at $p$, and $\mathbb{F}_p$ the field of $p$ elements. Let $\mathcal{X}$ be the class of $\mathbb{F}_p G$-modules obtained by reduction of simple $\mathbb{Q}G$-modules. Is it true that $\mathbb{Z}_{(p)} G$ is semiperfect if and only if each projective indecomposable $\mathbb{F}_p G$-module can be written as an $\mathbb{N}$-linear combination of modules in $\mathcal{X}$ inside the Grothendieck group of $\mathbb{F}_p G$?