Given a group word w in k variables, a finite group G and g ∈ G, we consider the number N w,G (g) of k-tuples g 1 , . . . , g k of elements of G such that w(g 1 , . . . , g k ) = g. In this work we study the functions N w,G for the class of nilpotent groups of nilpotency class 2. We show that, for the groups in this class, N w,G (1) ≥ |G| k−1 , an inequality that can be improved to N w,G (1) ≥ |G| k /|G w | (G w is the set of values taken by w on G) if G has odd order. This last result is explained by the fact that the functions N w,G are characters of G in this case. For groups of even order, all that can be said is that N w,G is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when N x n ,G is a character if G is a 2-group of nilpotency class 2. Finally we also address the (much harder) problem of studying if N w,G (g) ≥ |G| k−1 for g ∈ G w , proving that this is the case for the free p-groups of nilpotency class 2 and exponent p.

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,. .. , g k) ∈ G (k) such that w(g1,. .. , 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 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.

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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.