Word problems for finite nilpotent groups

Abstract

Let $w$ be a word in $k$ variables. For a finite nilpotent group~$G$, a conjecture of Amit states that $N_w(1)\ge|G|^{k-1}$, where for $g\in G$ the quantity $N_w(g)$ is the number of $k$-tuples $\mbox{$(g_1,\ldots,g_k)\in G^{(k)}$}$ such that $w(g_1,\ldots,g_k)=g$. Currently, this conjecture is known to be true for groups of nilpotency class $2$. Here we consider a generalized version of Amit's conjecture, which states that $N_w(g)\ge |G|^{k-1}$ for $g$ a $w$-value in~$G$, and prove that $N_w(g)\ge |G|^{k-2}$, for finite groups~$G$ of odd order and nilpotency class~$2$. If $w$ is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups~$G$ of nilpotency class~$2$. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite $p$-groups ($p$ a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.