Word problems for finite nilpotent groups

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Camina, Rachel D. 
Iñiguez, Ainhoa 
Thillaisundaram, Anitha  ORCID logo  https://orcid.org/0000-0001-8768-2590

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.


Funder: University of Lincoln

Article, Words, Amit’s conjecture, Rational words, Primary 20F10, Secondary 20D15
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Archiv der Mathematik
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Springer International Publishing