Word problems for finite nilpotent groups

Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that N-w(1) >= vertical bar G vertical bar(k-1), where for g is an element of G, the quantity N-w(g) is the number of k-tuples (g(1), ..., g(k)) is an element of G((k)) such that w(g(1), ..., 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) >= vertical bar G vertical bar(k-1) for g a w-value in G, and prove that N-w(g) = vertical bar G vertical bar(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.