Sylow synchronization conjecture
For every finite group , every prime dividing , and every Sylow -subgroup of , one has , where denotes the set of irreducible complex characters of whose degrees are not divisible by , and is the normalizer of in .
References
Primary source
Additional references
- Sylow synchronization in finite groups: the good case — arXiv — Hong Yi Huang, Francesca Lisi, Aluna Rizzoli, Luca Sabatini
Progress summary
The conjecture has new proofs for broad special cases, while an earlier report claims the full statement was proved, but that claim has not been independently verified and the latest summary still calls the general problem open.
The conjecture is the finite-group representation-counting statement comparing a group with an appropriate Sylow normalizer. It is associated with McKay, who proposed it in 1972.
Known results
- Isaacs proved the statement for a large class of finite groups.
- Isaacs, Navarro, and Malle reduced the remaining cases to particular groups of Lie type.
- Späth and Cabanes supplied the remaining correspondence required in the claimed full proof.
September 2026 good-case advance; claimed full proof reported in 2025
On September 23, 2026, Huang, Lisi, Rizzoli, and Sabatini reported a broad good-case theorem, including all symmetric and alternating groups; their summary says the general conjecture remains open. Separately, a February 19, 2025 report described Späth and Cabanes's claimed completion of the corresponding McKay conjecture for every finite group. That identification and proof remain unverified here.
Current status (as of September 2026): broad good-case and symmetric/alternating cases are reported as proved, the latest summary calls the general conjecture open, and a purported full McKay-conjecture resolution remains unverified.
Solutions 0
No solutions have been posted yet.