Structural Explanation of the McKay Conjecture

Definitions and mathematical statement

For a finite group $H$, let $\operatorname{Irr}(H)$ denote the set of irreducible ordinary complex characters of $H$. For a prime $\ell$, define \[ \operatorname{Irr}_{\ell'}(H)=\{\chi\in\operatorname{Irr}(H):\ell\nmid\chi(1)\}. \] Let $G$ be a finite group, let $\ell$ be a prime, and let $P$ be a Sylow $\ell$-subgroup of $G$; thus $|P|=\ell^a$, where $\ell^a$ is the largest power of $\ell$ dividing $|G|$. Define its normalizer by \[ N_G(P)=\{g\in G:gPg^{-1}=P\}. \] The equality referred to as the McKay conjecture in the supplied record is \[ \left|\operatorname{Irr}_{\ell'}(G)\right| = \left|\operatorname{Irr}_{\ell'}(N_G(P))\right| \qquad\text{for every finite group }G,\text{ every prime }\ell,\text{ and every Sylow }\ell\text{-subgroup }P. \] This includes $\ell\nmid |G|$, when $P=\{1\}$ and $N_G(P)=G$.

Source question. Give a structural explanation of why this equality should hold. The finite-group setting and ordinary complex-character notation above make explicit the standard interpretation of the supplied formulation and report context. The source asks an explanatory question, rather than specifying a particular construction or a formally defined criterion for an explanation. Consequently, the equality precisely identifies the phenomenon to be explained, but the intended meaning of a satisfactory structural explanation remains unspecified. Its presentation here does not independently assess its current mathematical status.

Short English statement

The question asks for a structural reason why a finite group and the normalizer of a Sylow $\ell$-subgroup should have equally many irreducible complex characters whose degrees are not divisible by $\ell$. The supplied record labels this explanatory question open; it does not specify what would count as a satisfactory explanation.

Sources

• Local UnsolvedMath registry, supplied source record, permanent ID $30002853$, title Structural Explanation of the McKay Conjecture, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Representations of Finite Groups (2016), Oberwolfach Reports, pp. 971--1027, DOI 10.4171/owr/2015/18. Source URL and supplied literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2015/18. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and says that the structural explanation remains unclear, treating it as an open explanatory question rather than a resolved mathematical assertion. This is supplied dated curation, not independently verified; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified, with the qualification that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is a registry annotation, not independent verification of this draft.