The Amit-Ashurst conjecture for finite metacyclic p-groups

TL;DR

This paper proves the Amit-Ashurst conjecture for finite p-groups with a cyclic maximal subgroup, confirming a probabilistic property of word maps in this class of groups.

Contribution

It establishes the validity of the Amit-Ashurst conjecture for a new class of finite p-groups, expanding the understanding of word maps in group theory.

Findings

The conjecture holds for finite p-groups with cyclic maximal subgroups.

The probability of an element in the image of a word map is either 0 or at least 1/|G|.

Supports the conjecture's applicability to broader classes of nilpotent groups.

Abstract

The Amit conjecture about word maps on finite nilpotent groups has been shown to hold for certain classes of groups. The generalised Amit conjecture says that the probability of an element occurring in the image of a word map on a finite nilpotent group G is either 0, or at least 1/|G|. Noting the work of Ashurst, we name the generalised Amit conjecture the Amit-Ashurst conjecture and show that the Amit-Ashurst conjecture holds for finite p-groups with a cyclic maximal subgroup.