Kourovka Notebook Problem 21.60

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $p$ be prime, $\mathbb Z_{(p)}=\{a/b\in\mathbb Q:p\nmid b\}$, and $\mathbb F_p=\mathbb Z/p\mathbb Z$. For each simple finite-dimensional $\mathbb QG$-module $V$, choose a $G$-stable full $\mathbb Z_{(p)}$-lattice $L\subset V$ and reduce it to the $\mathbb F_pG$-module $L/pL$. Let $\mathcal X$ consist of these reductions, for all such $V,L$. The Grothendieck group $G_0(\mathbb F_pG)$ is generated by classes $[M]$ of finite-dimensional modules, with $[M]=[M']+[M'']$ for every short exact sequence $0\to M'\to M\to M''\to0$. A projective indecomposable module is a nonzero projective module not expressible as a direct sum of two nonzero modules. A ring $R$ is semiperfect when $R/J(R)$ is semisimple Artinian and idempotents lift modulo its Jacobson radical $J(R)$. Here an $\mathbb N$-linear combination permits nonnegative integer coefficients, including zero.

Statement recorded in the supplied source.

For every finite group $G$ and prime $p$, is $\mathbb Z_{(p)}G$ semiperfect if and only if, for each projective indecomposable $\mathbb F_pG$-module $P$, there exist $X_1,\ldots,X_m\in\mathcal X$ and $a_1,\ldots,a_m\in\mathbb Z_{\geq0}$ with \[ [P]=\sum_{i=1}^{m}a_i[X_i]\quad\text{in }G_0(\mathbb F_pG)? \] The assertion is equality of Grothendieck classes, rather than an isomorphism of modules to that direct sum. [S2]

Short English statement

Is semiperfectness of the localized integral group algebra equivalent to a nonnegative reduction formula for every indecomposable projective modular representation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2569 (KOU-21.60). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 D. Johnston and D. Rumynin, Problem 21.60, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). This version records a negative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.