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The Picky Conjecture for finite groups

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Let GG be a finite group, let ℓ\ell be a prime, let P∈Syl⁡ℓ(G)P\in\operatorname{Syl}_\ell(G), and let x∈Px\in P be a picky ℓ\ell-element, meaning that xx lies in a unique Sylow ℓ\ell-subgroup of GG. For any finite group HH and y∈Hy\in H, write Irr⁡y(H)\operatorname{Irr}^y(H) for the set of complex irreducible characters of HH that take a nonzero value at yy.

Picky Conjecture. There exists a bijection

f:Irr⁡x(G)→Irr⁡x(N⁡G(P))f:\operatorname{Irr}^x(G)\to\operatorname{Irr}^x(\operatorname{N}_G(P))

such that, for every χ∈Irr⁡x(G)\chi\in\operatorname{Irr}^x(G),

χ(1)ℓ=f(χ)(1)ℓ\chi(1)_\ell=f(\chi)(1)_\ell

and

Q(χ(x))=Q(f(χ)(x)).\mathbb{Q}(\chi(x))=\mathbb{Q}(f(\chi)(x)).

We say that the Picky Conjecture holds for (G,ℓ,x)(G,\ell,x) when these degree and character-field conditions hold.

The conjecture proposes a character correspondence extending the McKay conjecture. The source paper proves it for all quasi-simple groups of Lie type in non-defining characteristic; its general status beyond those cases is not resolved here.

Progress summary

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The Picky Conjecture for finite groups

    Let pp be a prime, let GG be a finite group, and let P∈Syl⁡p(G)P\in\operatorname{Syl}_p(G). For x∈Px\in P, call xx picky if it lies in a unique Sylow pp-subgroup of GG. Define

    Irr⁡x(G)={χ∈Irr⁡(G)∣χ(x)≠0}.\operatorname{Irr}^x(G)=\{\chi\in\operatorname{Irr}(G)\mid \chi(x)\ne 0\}.

    Picky Conjecture. If x∈Px\in P is picky, then there exists a bijection

    Γ:Irr⁡x(G)⟶Irr⁡x(NG(P))\Gamma:\operatorname{Irr}^x(G)\longrightarrow\operatorname{Irr}^x(\mathbf{N}_G(P))

    satisfying (I)(I) Γ(χ)(1)p=χ(1)p\Gamma(\chi)(1)_p=\chi(1)_p for every χ∈Irr⁡x(G)\chi\in\operatorname{Irr}^x(G), and (II)(II) Q(Γ(χ)(x))=Q(χ(x))\mathbb{Q}(\Gamma(\chi)(x))=\mathbb{Q}(\chi(x)) for every χ∈Irr⁡x(G)\chi\in\operatorname{Irr}^x(G). This character-theoretic local-global conjecture is related to the McKay conjecture and was formulated in the cited source. It has been proved by Cabanes and Späth.

    source: Juan Martínez Madrid, “The Picky and Subnormalizer Conjectures for symmetric groups”, arXiv:2508.05180 (2026).

Sources & referencesView supporting material

Primary source

Gunter Malle and A. A. Schaeffer Fry, “The Picky Conjecture for groups of Lie type”, arXiv:2510.18397 (2025).

Additional references

2 papers in this index state this conjecture (2025). The statement above is taken from the most recent of them; the others are arXiv:2508.05180.

character-correspondencescharacter-theoryfinite-groupsmckay-conjecturerepresentation-theory

Solutions 0

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