Products of unipotent matrices of index 2 over division rings

Let D be a division ring. The first aim of this paper is to describe all unipotent matrices of index 2 in the general linear group GL n ( D ) of degree n and in the Vershik–Kerov group GL VK ( D ) . As a corollary, the subgroups generated by such matrices are investigated. The next aim is to seek a...

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Bibliographic Details
Published in:Acta mathematica Hungarica 2024-06, Vol.173 (1), p.74-100
Main Authors: Bien, M. H., Son, T. N., Thuy, P. T. T., Truong, L. Q.
Format: Article
Language:English
Subjects:
Commutators
Mathematics
Mathematics and Statistics
Rings (mathematics)
Subgroups
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Summary:Let D be a division ring. The first aim of this paper is to describe all unipotent matrices of index 2 in the general linear group GL n ( D ) of degree n and in the Vershik–Kerov group GL VK ( D ) . As a corollary, the subgroups generated by such matrices are investigated. The next aim is to seek a positive integer d such that every matrix in these groups is a product of at most d unipotent matrices of index 2. For example, we show that if every element in the derived subgroup D ′ of D ∗ = D \ { 0 } is a product of at most c commutators in D ∗ , then every matrix in GL n ( D ) (resp., GL VK ( D ) , which is a product of some unipotent matrices of index 2, can be written as a product of at most 4+3 c (resp.,5 + 3 c ) of unipotent matrices of index 2 in GL n ( D ) (resp., GL VK ( D ) ) .
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-024-01427-w