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ISSN 1547-7371(e) ISSN 1061-0022(p)

On (2,3)-generation of matrix groups over the ring of integers

Author(s): M. A. Vsemirnov
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 6.
Journal: St. Petersburg Math. J. 19 (2008), 883-910.
MSC (2000): Primary 20G30; Secondary 20F05, 20C12
Posted: August 21, 2008
MathSciNet review: 2411638
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Abstract | References | Similar articles | Additional information

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:

1.: L. Di Martino and N. Vavilov, -generation of $\mathrm{SL}(n,q)$ . I. Cases , Comm. Algebra 22 (1994), no. 4, 1321-1347. MR 1261262 (95f:20076)
2.: -, -generation of $\mathrm{SL}(n,q)$ . II. Cases $n\geq 8$ , Comm. Algebra 24 (1996), 487-515. MR 1373489 (97e:20067)
3.: A. J. Hahn and O. T. O'Meara, The classical groups and -theory, Grundlehren Math. Wiss., Bd. 291, Springer-Verlag, Berlin, 1989. MR 1007302 (90i:20002)
4.: M. W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the -generation problem, Ann. of Math. (2) 144 (1996), 77-125. MR 1405944 (97e:20106a)
5.: A. Yu. Luzgarëv and I. M. Pevzner, Some facts about the life of $\mathrm{GL}(5,\mathbb{Z})$ , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 305 (2003), 153-162; English transl., J. Math. Sci. (N. Y.) 130 (2005), no. 3, 4729-4733. MR 2033620
6.: G. A. Miller, On the groups generated by two operators, Bull. Amer. Math. Soc. 7 (1901), no. 10, 424-426. MR 1557828
7.: Ya. N. Nuzhin, On a question of M. Conder, Mat. Zametki 70 (2001), no. 1, 79-87; English transl., Math. Notes 70 (2001), no. 1-2, 71-78. MR 1883773 (2002m:20076)
8.: P. Sanchini and M. C. Tamburini, Constructive -generation: A permutational approach, Rend. Sem. Mat. Fis. Milano 64 (1994), 141-158 (1996). MR 1397469 (97i:20040)
9.: M. C. Tamburini and J. S. Wilson, On the -generation of some classical groups. II, J. Algebra 176 (1995), 667-680. MR 1351631 (96h:20085)
10.: M. C. Tamburini, J. S. Wilson, and N. Gavioli, On the -generation of some classical groups. I, J. Algebra 168 (1994), 353-370. MR 1289105 (95f:20075)
11.: M. C. Tamburini and P. 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. MR 1797048 (2001i:20097)
12.: M. A. Vsemirnov, Is the group $\mathrm{SL}(6,\mathbb{Z})$ -generated? Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006), 101-130; English transl., J. Math. Sci. (N. Y.) 140 (2007), no. 5, 660-675. MR 2253569 (2008e:20074)
13.: -, The group $\mathrm{GL}(6,\mathbb{Z})$ is -generated, J. Group Theory 10 (2007), no. 4, 425-430. MR 2334749 (2008f:20118)

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Additional 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

DOI: 10.1090/S1061-0022-08-01026-1
PII: S 1061-0022(08)01026-1
Keywords: $(2,3)$-generation, linear groups
Received by editor(s): 10/AUG/2007
Posted: 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)
Dedicated: To the centenary of D. K. Faddeev's birth
Copyright of article: Copyright 2008, American Mathematical Society

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