Hexagon

math.GR — Group Theory

Explicit Finite $p$ -Group Counterexamples to the Amit--Ashurst Conjecture

Contributed by Shihao Wang, Zhiwei Zheng

For every prime $p$ , we construct a finite $p$ -group $G$ of order $p^{498}$ and nilpotency class two for which the word $w=\prod_{j=1}^{12}[x_{2j-1},x_{2j}]$ admits a value $g\ne1$ whose fibre has size strictly less than $|G|^{23}$ . This gives counterexamples to the Amit--Ashurst conjecture for every prime. We determine the exact fibre size by reducing the word equation to matrix equations over $\mathbb{F}_p$ and counting their solutions.

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Oct 26, 2026PDF ↗View work →

hexagon:2610.00529v1

math.GR — Group Theory

A Counterexample to Huppert's Conjecture

Contributed by Shihao Wang, Zhiwei Zheng

Let $S = \mathrm{PSL}_4(5)$ and let $G$ be the unique subgroup of $\mathrm{PGL}_4(5)$ containing $S$ with $[G:S] = 2$ . We prove that $G$ and $S$ have the same set of irreducible complex character degrees. Meanwhile, $G$ is centerless and is not isomorphic to $S \times A$ for any finite abelian group $A$ . This provides a counterexample to Huppert's conjecture.

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Oct 26, 2026PDF ↗View work →

Submitted works

Explicit Finite ppp-Group Counterexamples to the Amit--Ashurst Conjecture

A Counterexample to Huppert's Conjecture

Explicit Finite $p$ -Group Counterexamples to the Amit--Ashurst Conjecture