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

Shihao Wang; Zhiwei Zheng

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

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Contributed by Shihao Wang ↗, Zhiwei Zheng ↗

Version 1 / Oct 26, 2026 / CC BY-SA 4.0

Abstract

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.

References

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Version history

v1Submitted by Shihao WangInitial depositCurrentOct 26, 2026