McKay Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite group $G$, let $\operatorname{Irr}(G)$ denote the irreducible complex characters, one per equivalence class of irreducible finite-dimensional complex representations. Their degrees are $\chi(1)$. For a prime $p$, put \[ \operatorname{Irr}_{p'}(G) =\{\chi\in\operatorname{Irr}(G):p\nmid\chi(1)\}. \] A Sylow $p$-subgroup $P\le G$ has order equal to the largest power of $p$ dividing $|G|$. Its normalizer is $N_G(P)=\{g\in G:gPg^{-1}=P\}$.

Statement recorded in the supplied source.

For every finite group $G$, every prime $p$, and every Sylow $p$-subgroup $P\le G$, is \[ |\operatorname{Irr}_{p'}(G)| =|\operatorname{Irr}_{p'}(N_G(P))|? \] This compares the numbers of characters of degree prime to $p$; it does not require an additional Galois-equivariance property. [S2]

Short English statement

Does a finite group have as many irreducible complex characters of degree prime to p as the normalizer of a Sylow p-subgroup?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1114 (ALG-014). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Lucas Ruhstorfer and A. A. Schaeffer Fry, The McKay--Navarro conjecture for the prime 2, arXiv:2211.14237, revised 2025, introduction, first paragraph stating the ordinary McKay conjecture. https://arxiv.org/abs/2211.14237.