Kourovka Notebook Problem 21.91

Definitions and mathematical statement

Let $G$ be a finite group, let $p$ be a prime, and let $k$ be an algebraically closed field of characteristic $p$. A simple $kG$-module is a nonzero finite-dimensional $k$-vector space equipped with a linear $G$-action and having no proper nonzero invariant subspace. Choose one representative $V_1,\ldots,V_t$ from each isomorphism class of simple $kG$-modules, and put $d_i=\dim_kV_i$.

The irreducible $p$-Brauer characters of $G$ correspond to these modules; their degrees, meaning their values at the identity, are precisely $d_1,\ldots,d_t$. Write \[ |G|=p^a m,\qquad a\ge0,\quad p\nmid m, \qquad |G|_{p'}=m. \] The conjecture asks whether every finite $G$ and every prime $p$ satisfy \[ \sum_{i=1}^{t}d_i^2\ \ge\ |G|_{p'}. \] The sum includes every isomorphism class of simple module once, even when different modules have the same dimension. It does not weight a degree by the multiplicity of that module in the regular representation. Equivalently the sum is the dimension of the semisimple quotient $kG/J(kG)$, where $J(kG)$ is the Jacobson radical of the group algebra. This modular formulation specifies the same inequality without needing choices of lifts used to define Brauer-character values away from the identity.

Short English statement

Is the sum of squared irreducible modular character degrees always at least the prime-to-$p$ part of the group order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2600 (KOU-21.91), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.91. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.