Math Conjectures

Brauer’s k(B)k(B)-Conjecture

OPENMajorConjectureProposed 1946 · Full conjecture

Canonical statement

Let GG be a finite group, let pp be a prime, and fix a splitting pp-modular system for GG. Let BB be a pp-block of GG, meaning a two-sided summand of the modular group algebra cut out by a primitive central idempotent; let Irr⁡(B)\operatorname{Irr}(B) be the set of irreducible ordinary complex characters belonging to BB, and let D≤GD\le G be a defect group of BB. Then
k(B):=#Irr⁡(B)≤∣D∣. k(B):=\#\operatorname{Irr}(B)\le |D|.
View source LaTeX
Let \(G\) be a finite group, let \(p\) be a prime, and fix a splitting \(p\)-modular system for \(G\). Let \(B\) be a \(p\)-block of \(G\), meaning a two-sided summand of the modular group algebra cut out by a primitive central idempotent; let \(\operatorname{Irr}(B)\) be the set of irreducible ordinary complex characters belonging to \(B\), and let \(D\le G\) be a defect group of \(B\). Then
\[
  k(B):=\#\operatorname{Irr}(B)\le |D|.
\]
ALG-MODREP-019:[Brauer1946Blocks][Sambale2014Blocks][Malle2018BrauerKB][Sambale2019BoundingKB]
Algebra and representation theoryNumber theory

Notes

The inequality is known for many classes of groups and blocks, including numerous cases with abelian or otherwise restricted defect groups, and reductions constrain a minimal counterexample. No proof covers arbitrary finite groups and arbitrary blocks.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.