# Nilpotency and the number of word maps of a finite group

## TL;DR

This paper characterizes the nilpotency of finite groups by analyzing the asymptotic growth of the number of word-induced functions as the number of variables increases.

## Contribution

It introduces a novel characterization of nilpotency in finite groups based on the growth rate of word maps.

## Key findings

- Nilpotent groups exhibit a specific growth pattern in the number of word maps.
- The growth rate of $ ext{Ω}_d(G)$ distinguishes nilpotent groups from non-nilpotent ones.
- Asymptotic analysis provides a new tool for understanding group structure.

## Abstract

For a finite group $G$ and a non-negative integer $d$, denote by $\Omega_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that nilpotency of $G$ can be characterized through the asymptotic growth rate of $\Omega_d(G)$ as $d\to\infty$.