On (2,3)-generation of matrix groups over the ring of integers
HTML articles powered by AMS MathViewer
- by
M. A. Vsemirnov
Translated by: the author - St. Petersburg Math. J. 19 (2008), 883-910
- DOI: https://doi.org/10.1090/S1061-0022-08-01026-1
- Published electronically: August 21, 2008
- PDF | Request permission
Abstract:
The groups $\mathrm {GL}(5,\mathbb {Z})$, $\mathrm {SL}(5,\mathbb {Z})$, $\mathrm {SL}(6,\mathbb {Z})$, $\mathrm {GL}(7,\mathbb {Z})$ and $\mathrm {SL}(7,\mathbb {Z})$ are (2,3)-generated.References
- Lino Di Martino and Nikolai Vavilov, $(2,3)$-generation of $\textrm {SL}(n,q)$. I. Cases $n=5,6,7$, Comm. Algebra 22 (1994), no. 4, 1321–1347. MR 1261262, DOI 10.1080/00927879408824908
- Lino Di Martino and Nikolai Vavilov, $(2,3)$-generation of $\textrm {SL}(n,q)$. II. Cases $n\geq 8$, Comm. Algebra 24 (1996), no. 2, 487–515. MR 1373489, DOI 10.1080/00927879608825582
- Alexander J. Hahn and O. Timothy O’Meara, The classical groups and $K$-theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302, DOI 10.1007/978-3-662-13152-7
- Martin W. Liebeck and Aner Shalev, Classical groups, probabilistic methods, and the $(2,3)$-generation problem, Ann. of Math. (2) 144 (1996), no. 1, 77–125. MR 1405944, DOI 10.2307/2118584
- A. Yu. Luzgarëv and I. M. Pevzner, Some facts about the life of $\textrm {GL}(5,{\Bbb Z})$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 305 (2003), no. Vopr. Teor. Predst. Algebr. i Grupp. 10, 153–162, 240 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 130 (2005), no. 3, 4729–4733. MR 2033620, DOI 10.1007/s10958-005-0368-8
- G. A. Miller, On the groups generated by two operators, Bull. Amer. Math. Soc. 7 (1901), no. 10, 424–426. MR 1557828, DOI 10.1090/S0002-9904-1901-00826-9
- Ya. N. Nuzhin, On a question of M. Conder, Mat. Zametki 70 (2001), no. 1, 79–87 (Russian, with Russian summary); English transl., Math. Notes 70 (2001), no. 1-2, 71–78. MR 1883773, DOI 10.1023/A:1010273918392
- Paolo Sanchini and M. Chiara Tamburini, Constructive $(2,3)$-generation: a permutational approach, Rend. Sem. Mat. Fis. Milano 64 (1994), 141–158 (1996). MR 1397469, DOI 10.1007/BF02925196
- M. Chiara Tamburini and John S. Wilson, On the $(2,3)$-generation of some classical groups. II, J. Algebra 176 (1995), no. 2, 667–680. MR 1351631, DOI 10.1006/jabr.1995.1266
- M. Chiara Tamburini, John S. Wilson, and Norberto Gavioli, On the $(2,3)$-generation of some classical groups. I, J. Algebra 168 (1994), no. 1, 353–370. MR 1289105, DOI 10.1006/jabr.1994.1234
- M. Chiara Tamburini and Paola Zucca, On a question of M. Conder, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 1, 5–7 (English, with English and Italian summaries). MR 1797048
- M. A. Vsemirnov, Is the group $\textrm {SL}(6,{\Bbb Z})$ $(2,3)$-generated?, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 13, 101–130, 272 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 5, 660–675. MR 2253569, DOI 10.1007/s10958-007-0006-8
- Maxim Vsemirnov, The group $\textrm {GL}(6,\Bbb Z)$ is $(2,3)$-generated, J. Group Theory 10 (2007), no. 4, 425–430. MR 2334749, DOI 10.1515/JGT.2007.033
Bibliographic Information
- M. A. Vsemirnov
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
- Email: vsemir@pdmi.ras.ru
- Received by editor(s): August 10, 2007
- Published electronically: August 21, 2008
- Additional Notes: Supported by the RAS program of fundamental research “Modern problems of theoretical mathematics” and by the “Scientific schools” program (grant no. NSh-8464-2006-1)
- © Copyright 2008 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 883-910
- MSC (2000): Primary 20G30; Secondary 20F05, 20C12
- DOI: https://doi.org/10.1090/S1061-0022-08-01026-1
- MathSciNet review: 2411638
Dedicated: To the centenary of D. K. Faddeev’s birth