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.