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

IF 0.6 4区数学 Q3 MATHEMATICS Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s0001434624030015

M. A. Vsemirnov, R. I. Gvozdev, Ya. N. Nuzhin, T. B. Shaipova

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Abstract

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 unknown 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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论 $$\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$$ 和 $$\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$$ 群的生成--三个旋回其中两个相交.二

摘要我们完成了对高斯整数环上的$\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$特殊线性群和投影特殊线性群$\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$的特殊线性群的生成三胞胎的求解，其中两个三胞胎是相通的。答案只有 $\mathrm{SL}_5$、$\mathrm{PSL}_6$和$\mathrm{SL}_{10}$是未知的。我们明确地指出了这三种情况下渐开线的生成三元组，并在证明中大量使用了计算机计算。考虑到所考虑问题的已知结果，我们得到了以下两个陈述。组$\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$（分别是$\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$是由三个渐开线生成的，其中两个渐开线只有在$n\geq 5$ 和$n\neq 6$ （分别是如果$n\geq 5$ ）的情况下才会换向。）

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.