Navarro’s conjecture on Brauer characters under coprime actions

Let $G$ and $A$ be finite groups such that $A$ acts on $G$ by automorphisms and $\gcd(|A|,|G|)=1$ . For every prime $p$ , the number of $A$ -invariant irreducible $p$ -Brauer characters of $G$ equals the number of irreducible $p$ -Brauer characters of the centralizer $\mathrm{C}_G(A)$ ; that is, $\bigl|\operatorname{IBr}_p(G)^A\bigr|=\bigl|\operatorname{IBr}_p(\mathrm{C}_G(A))\bigr|$ .

References

Primary source

arXiv

Additional references

The Brauer A(∞) condition and Navarro's conjecture on Brauer characters under coprime actions — arXiv — Zhicheng Feng, Lin Wu

finite-group-theory representation-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A 2026 paper claims to settle the conjecture for all primes greater than three, while the cases two and three remain open.

Navarro’s conjecture predicts that, for a coprime action of $A$ on $G$ , the number of $A$ -invariant irreducible Brauer characters of $G$ equals the number of irreducible Brauer characters of the centralizer $\mathrm{C}_G(A)$ . Uno first posed the problem; later work treated the $p$ -solvable case and reduced the general problem to quasi-simple groups.

Known results

Wolf and Graves proved the conjecture in the $p$ -solvable case.
Späth and Vallejo reduced the general conjecture to quasi-simple groups in 2016.

2026 Brauer $A(\infty)$ development

Zhicheng Feng and Lin Wu claim that their arXiv paper proves the Brauer $A(\infty)$ condition for good non-defining primes and hence establishes Navarro’s equality for every $p>3$ . The paper states that only $p=2$ and $p=3$ remain open, but this claimed advance has not been independently verified.

Current status (as of September 2026): The conjecture is claimed for $p>3$ ; the cases $p=2$ and $p=3$ remain open, and the new proof is unverified.

Sources

No solutions have been posted yet.