Camina, Rachel, Iniguez, Ainhoa and Thillaisundaram, Anitha (2020) Word problems for finite nilpotent groups. Archiv der Mathematik, 115 (6). pp. 599609. ISSN 0003889X
Full content URL: https://doi.org/10.1007/s0001302001504w
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Abstract
Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1)\geG^{k1}$, where for $g\in G$ 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 $N_w(g)\ge G^{k1}$ for $g$ a $w$value in $G$, and prove that $N_w(g)\ge G^{k2}$, 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.
Keywords:  Words, Amit's conjecture, rational words 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  41522 
Deposited On:  20 Jul 2020 15:38 
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