Word Images and Their Impostors in Finite Nilpotent Groups

TL;DR

This paper investigates the structure of word maps in finite nilpotent groups, revealing that automorphism invariant subsets not arising from words can vastly outnumber actual word images, contrasting with known results for simple groups.

Contribution

It introduces the concept of word image impostors in finite nilpotent groups and constructs a minimal 2-exhaustive set of word maps for class 2 nilpotent groups.

Findings

Automorphism invariant subsets not from words can be arbitrarily numerous.

Constructed a minimal 2-exhaustive set of word maps for class 2 nilpotent groups.

Contrasts behavior with finite simple groups regarding word images.

Abstract

It was shown by Lubotzky in 2014 that automorphism invariant subsets of finite simple groups which contain identity are always word images. In this article, we study word maps on finite nilpotent groups and show that for arbitrary finite groups, the number of automorphism invariant subsets containing identity which are not word images, referred to as word image impostors, may be arbitrarily larger than the number of actual word images. In the course of it, we construct a $2$-exhaustive set of word maps on nilpotent groups of class $2$ and demonstrate its minimality in some cases.