On the Generation of the Groups $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$ by Three Involutions Two of Which Commute. II : научное издание

Описание

Тип публикации: статья из журнала

Год издания: 2024

Идентификатор DOI: 10.1134/S0001434624030015

Ключевые слова: special and projective special linear groups, ring of Gaussian integers, generating triplet of involutions

Аннотация: We complete the solution of the problem on the existence of generating triplets of involutions two of which commute for the special linear group $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and the projective special linear group $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$ over the ring of Gaussian integers. The answer has only been unknowПоказать полностьюn for $\mathrm{SL}_5$, $\mathrm{PSL}_6$, and $\mathrm{SL}_{10}$. We explicitly indicate the generating triples of involutions in these three cases, and we make a significant use of computer calculations in the proof. Taking into account the known results for the problem under consideration, as a consequence, we obtain the following two statements. The group $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ (respectively, $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$) is generated by three involutions two of which commute if and only if $n\geq 5$ and $n\neq 6$ (respectively, if $n\geq 5$).

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Издание

Журнал: Mathematical Notes

Выпуск журнала: Т.115, 3-4

Номера страниц: 289-300

ISSN журнала: 00014346

Место издания: Moscow

Издатель: Pleiades Publishing, Ltd.

Персоны

  • Vsemirnov M.A. (St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences)
  • Gvozdev R.I. (Siberian Federal University)
  • Nuzhin Ya.N. (Siberian Federal University)
  • Shaipova T.B. (Krasnoyarsk Scientific Center of Siberian Branch of Russian Academy of Sciences)

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