Word problems for finite nilpotent groups

Abstract

Abstract: Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that Nw(1)≥|G|k-1, where for g∈G, the quantity Nw(g) is the number of k-tuples (g1, …, gk)∈G(k) such that w(g1, …, gk)=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 Nw(g)≥|G|k-1 for g a w-value in G, and prove that Nw(g)≥|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.