{"id":"hexagon/2610.00563","pdf":"https://hexagonmath.org/pdf/2610.00563v1","url":"https://hexagonmath.org/abs/2610.00563v1","tags":["msc:20F10","msc:20D15"],"score":0,"title":"Explicit Finite $p$-Group Counterexamples to the Amit--Ashurst Conjecture","domain":"math.GR","origin":"ai-generated","source":"hexagon","authors":["Shihao Wang","Zhiwei Zheng"],"reviews":0,"subject":"math.GR","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}