Moreto’s conjecture on Sylow-subgroup character counts

Moreto’s conjecture asserts that, for a finite group $G$, prime $p$, and Sylow $p$-subgroup $P$, $n$ irreducible characters with $p$-divisible degrees force every subgroup chain from $N_G(P)$ to $G$ to have length at most $n$.

Known results

For $|\mathrm{Irr}_p(G)|=1$, $N_G(P)$ is maximal in $G$.

Counterexample report, September 22, 2026

Gang Chen and Wenhua Zhao report a counterexample with $p=3$, $|\mathrm{Irr}_p(G)|=2$, and $l_p(G)=3$, using a group of order $2^6\cdot3\cdot7$. Thus the conjectured bound fails. They also announce forthcoming positive results for supersolvable groups and some solvable groups. The paper says the construction was found with help from OpenAI’s GPT-6.

Current status (as of September 2026): A counterexample is claimed, but independent verification is outstanding; the conjecture is not established as false until that claim is checked.