Words and characters in finite p-groups

TL;DR

This paper investigates the functions counting solutions to group equations in finite nilpotent groups of class 2, revealing character properties and bounds, especially for groups of odd order and free p-groups.

Contribution

It characterizes when these counting functions are characters in nilpotent groups of class 2 and addresses bounds for solutions in free p-groups of the same class.

Findings

N_{w,G}(1) ; |G|^{k-1}

N_{w,G} is a character for odd order groups

Solution counts are bounded below in free p-groups

Abstract

Given a group word $w$ in $k$ variables, a finite group $G$ and $g\in G$, we consider the number $N_{w,G}(g)$ of $k$-tuples $g_1,\dots ,g_k$ of elements of $G$ such that $w(g_1,\dots ,g_k)=g$. In this work we study the functions $N_{w,G}$ for the class of nilpotent groups of nilpotency class $2$. We show that, for the groups in this class, $N_{w,G}(1)\geq |G|^{k-1}$, an inequality that can be improved to $N_{w,G}(1)\geq |G|^k/|G_w|$ ($G_w$ is the set of values taken by $w$ on $G$) if $G$ has odd order. This last result is explained by the fact that the functions $N_{w,G}$ are characters of $G$ in this case. For groups of even order, all that can be said is that $N_{w,G}$ is a generalized character, something that is false in general for groups of nilpotency class greater than $2$. We characterize group theoretically when $N_{x^n,G}$ is a character if $G$ is a $2$-group of nilpotency class $2$. Finally we also address the (much harder) problem of studying if $N_{w,G}(g)\geq |G|^{ k-1}$ for $g\in G_w$, proving that this is the case for the free $p$-groups of nilpotency class $2$ and exponent $p$.