Péterfalvi’s p=3 Glauberman–Norton group question

Péterfalvi’s remaining p=3 question asks whether Glauberman–Norton Hypothesis (B) can hold. For prime q≠2 it requires an embedded additive-F₍₃^q₎ semidirect product with norm-one units, together with a finite abelian 3′-subgroup and the specified conjugation-normalization relations.

Why it matters and what counts as progress

Why it matters: The p≥5 obstruction and p=2 examples leave p=3 as the unresolved boundary case of this group-theoretic hypothesis.

What counts as progress: A construction satisfying Hypothesis (B), or a nonexistence proof for p=3 covering every allowed q and both finite and infinite ambient groups.

How the problem developed

1993 · Open question [1][2]
G. Glauberman and S. P. Norton (question attributed to Thomas Péterfalvi) Ended their paper by asking whether Proposition 9's hypothesis can hold for p=3.
27 Sept 2026 · Lean-checked resolution [2]
Yawara Ishida Proved that no finite or infinite group satisfies the stated hypothesis for p=3 when the prime q does not divide 2; also verified the cited p=2 example.

AI contribution

Claude-led proof answers Péterfalvi’s p=3 question

For p=3 and every prime q≠2, no finite or infinite group G satisfies Glauberman–Norton Hypothesis (B): it would need an embedding of H=(additive F_{3^q})⋊(norm-one units), a finite abelian subgroup Q of order prime to 3 and y∈Q, with the embedded prime-field line normalizing Q and its y-conjugate normalizing the embedded U.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4/Mathlib kernel, Comparator, NanoDa and con-ron)

Human understandingFew