The best possible Higman function: is $\chi(p)=(p^2-1)/4$? (settle $p=11$) — Kourovka 6.21
Statement
G. Higman proved that for every prime $p$ there exists a natural number $\chi(p)$ such that the nilpotency class of any finite group admitting a fixed-point-free (regular) automorphism of order $p$ does not exceed $\chi(p)$, and he showed $\chi(p)\ge (p^2-1)/4$ for every such Higman function $\chi$. Find the best possible Higman function. Is it the function given by $\chi(p)=(p^2-1)/4$ for $p>2$ and $\chi(2)=1$? This is known to be true for $p\le 7$; it is open for every prime $p\ge 11$.
Acceptance. FULLY RESOLVES: a proof that the best possible Higman function is chi(p)=(p^2-1)/4 for all p>2 (with chi(2)=1), or the exhibition of a prime for which the best bound differs from (p^2-1)/4. ADVANCES: settle the smallest open prime p=11 — a machine-checkable certificate (Gröbner-basis / linear-algebra computation over F_11 in the free Z/11-graded nilpotent Lie ring with the scalar-acting automorphism) that every homogeneous Lie word of weight greater than (11^2-1)/4 = 30 vanishes, paired with the known Higman extremal example attaining class 30 — thereby proving chi(11)=30; likewise any single new prime p >= 11 is an independent, citable advance. All computations must ship re-runnable code.
Background
Source: The Kourovka Notebook, No. 21, arXiv:1401.0300 (v44, June 2026), Problem 6.21, posed by V. D. Mazurov (6th Issue, 1978), formalizing Higman's 1957 result (J. London Math. Soc. 32 (1957) 321–334). The matching lower bound chi(p) >= (p^2-1)/4 comes with an extremal example (Higman 1957). The only general upper bound, Kreknin–Kostrikin, is doubly exponential in p (about (p-1)^(2^(p-1))), astronomically far from the conjectured quadratic; the exact value is settled only for p <= 7. The problem reduces, via the associated Z/p-graded F_p Lie ring (the automorphism acting as the scalar zeta^i on the i-th graded component), to a finite nilpotent-Lie-ring vanishing computation, prime by prime. Adjacent recent work (Khukhro et al. on Frobenius groups of automorphisms, e.g. arXiv:1806.05529) does not settle the exact function. Vetted open as of 2026-07-06: multiple 2020–2026 sources describe it as open for p >= 11, and no preprint resolves p=11 or gives a uniform proof. Fidelity note: the notebook writes the conjectured value as (p^2-1)/4 with no ceiling brackets — this is already an integer for odd p (since p^2 = 1 mod 8), so it agrees with the ceil((p^2-1)/4) form seen elsewhere.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | The Kourovka Notebook No. 21 (v44, June 2026) — Problem 6.21 (V. D. Mazurov) | arxiv |
| REF-02 | Khukhro et al., Frobenius groups of automorphisms (adjacent Lie-ring methods) | arxiv |
Investigations · 0
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