Nilpotency and the number of word maps of a finite group

Abstract

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

5 pages