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) ≥ |G| k−1 , where N w (1) is the number of k-tuples (g 1 , . . . , g k ) ∈ G (k) such that w(g 1 , . . . , g k ) = 1. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit's conjecture, and prove that N w (g) ≥ |G| k−2 , where g is a w-value in G, for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that N w (g) ≥ |G|, where g is a w-value in G for finite groups G of nilpotency class 2. In addition, for p a prime, we show that finite p-groups G, with two distinct irreducible complex character degrees, satisfy the generalized Amit conjecture for wordsFinally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.