Word problems for finite nilpotent groups
(2020) In Archiv der Mathematik 115. p.599-609- 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... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/50e8ef7b-f793-4812-bf39-ca8d5594cb44
- author
- Camina, Rachel ; Iniquez, Ainhoa and Thillaisundaram, Anitha LU
- publishing date
- 2020
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Archiv der Mathematik
- volume
- 115
- pages
- 599 - 609
- publisher
- Birkhäuser
- external identifiers
-
- scopus:85088101701
- ISSN
- 1420-8938
- DOI
- 10.1007/s00013-020-01504-w
- language
- English
- LU publication?
- no
- id
- 50e8ef7b-f793-4812-bf39-ca8d5594cb44
- date added to LUP
- 2024-06-07 14:23:26
- date last changed
- 2025-04-04 14:50:46
@article{50e8ef7b-f793-4812-bf39-ca8d5594cb44,
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.}},
author = {{Camina, Rachel and Iniquez, Ainhoa and Thillaisundaram, Anitha}},
issn = {{1420-8938}},
language = {{eng}},
pages = {{599--609}},
publisher = {{Birkhäuser}},
series = {{Archiv der Mathematik}},
title = {{Word problems for finite nilpotent groups}},
url = {{http://dx.doi.org/10.1007/s00013-020-01504-w}},
doi = {{10.1007/s00013-020-01504-w}},
volume = {{115}},
year = {{2020}},
}