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 gin G, the quantity N_w(g) is the number of k-tuples (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.
Highlights
A word w in k variables x1, . . . , xk is an element in the free group Fk on x1, . . . , xk
We denote by Gw the set of word values of w in G, i.e., the set of elements g ∈ G such that the equation w = g has a solution in G(k), the direct product of k copies of G
Iniguez and Sangroniz [13] proved that for any finite group G of nilpotency class 2 and any word w, the function Nw is a generalized character of G, that is, a Z-linear combination of irreducible characters
Summary
A word w in k variables x1, . . . , xk is an element in the free group Fk on x1, . . . , xk. Iniguez and Sangroniz [13] proved that for any finite group G of nilpotency class 2 and any word w, the function Nw is a generalized character of G, that is, a Z-linear combination of irreducible characters. If G is a finite p-group of nilpotency class 2 with p odd and w any word, Nw is a character of G.
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