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Products of unipotent matrices of index 2 over division rings

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Abstract

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 \(\mathrm {GL}_n(D)\) of degree n and in the Vershik–Kerov group \(\mathrm{GL} _{\rm VK}(D)\). 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 \(D^*=D\backslash \{0\}\) is a product of at most c commutators in \(D^*\), then every matrix in \(\mathrm{GL}_n(D)\) (resp., \(\mathrm{GL} _{\rm VK}(D)\), which is a product of some unipotent matrices of index 2, can be written as a product of at most 4+3c (resp.,5 + 3c) of unipotent matrices of index 2 in \(\mathrm{GL}_n(D)\) (resp., \(\mathrm{GL}_{\rm VK}(D))\).

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Acknowledgement

The authors would like to express their sincere appreciation for the referee’s careful reading and invaluable comments to improve this paper.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

    M. H. Bien, T. N. Son & L. Q. Truong

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    M. H. Bien, T. N. Son & L. Q. Truong

  3. Department of Mathematics and Informatics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    P. T. T. Thuy & L. Q. Truong

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  1. M. H. Bien
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  2. T. N. Son
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  3. P. T. T. Thuy
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  4. L. Q. Truong
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Correspondence to M. H. Bien.

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The first author was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2023.18.

The second author was funded by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), code VINIF.2023.TS.102.

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Bien, M.H., Son, T.N., Thuy, P.T.T. et al. Products of unipotent matrices of index 2 over division rings. Acta Math. Hungar. (2024). https://doi.org/10.1007/s10474-024-01427-w

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  • DOI: https://doi.org/10.1007/s10474-024-01427-w

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