×

Generation of finite simple groups by an involution and an element of prime order. (English) Zbl 1376.20018

Let \(G\) be a finite non-abelian simple group. The main result of this paper is that \(G\) is generated by an involution and some element of prime order \(p\), where \(p\) is dependent on \(G\). Many papers dealt with \(p = 3\). Not all simple groups are generated by an involution and an element of order three, but most of the sporadic and the alternating groups are generated in such a way and also the exceptional groups besides of course \({}^2B_2(q)\). The paper first deals with the remaining alternating groups and sporadic groups. Then, the author shows that for the classical groups \(\mathrm{Cl}_n(q)\) one can use a Zsigmondy prime \(p\), where \(p\) divides \(q^{2n}-1\) or \(q^{2n-2}-1\) for the unitary groups depending on \(n\) even or odd and in the remaining cases \(p\) divides \(q^n-1\), \(q^{n-1} -1 \) or \(q^{n-2}-1\), depending on the particular group and on \(q\).

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D05 Finite simple groups and their classification
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F05 Generators, relations, and presentations of groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math., 76, 469-514 (1984) · Zbl 0537.20023
[2] Aschbacher, M.; Guralnick, R., Some applications of the first cohomology group, J. Algebra, 90, 446-460 (1984) · Zbl 0554.20017
[3] Aschbacher, M.; Seitz, G. M., Involutions in Chevalley groups over fields of even order, Nagoya Math. J., 63, 1-91 (1976) · Zbl 0359.20014
[4] Bray, J. N.; Holt, D. F.; Roney-Dougal, C. M., The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Math. Soc. Lecture Note Ser., vol. 407 (2013), Cambridge Univ. Press · Zbl 1303.20053
[5] Burkhardt, R., Die Zerlegungsmatrizen der Gruppen \(P S L(2, p^f)\), J. Algebra, 40, 75-96 (1976) · Zbl 0334.20008
[6] Buturlakin, A. A.; Grechkoseeva, M. A., The cyclic structure of maximal tori of the finite classical groups, Algebra Logika, 46, 73-89 (2007) · Zbl 1155.20047
[7] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., An ATLAS of Finite Groups (1985), Clarendon Press: Clarendon Press Oxford · Zbl 0568.20001
[8] Enomoto, H., The characters of the finite symplectic group \(S p(4, q), q = 2^f\), Osaka J. Math., 9, 75-94 (1972) · Zbl 0254.20005
[9] Evans, M. J., A note on two-generator groups, Rocky Mountain J. Math., 17, 887-889 (1987) · Zbl 0688.20012
[10] Gorenstein, D.; Lyons, R.; Solomon, R., The Classification of the Finite Simple Groups, Number 3, Math. Surveys Monogr., vol. 40 (1998), Am. Math. Soc. · Zbl 0890.20012
[11] Guralnick, R. M.; Kantor, W. M., Probabilistic generation of finite simple groups. Special issue in honor of Helmut Wielandt, J. Algebra, 234, 743-792 (2000) · Zbl 0973.20012
[12] Guralnick, R.; Magaard, K.; Tiep, J. Saxl. P.H., Cross characteristic representations of symplectic and unitary groups, J. Algebra, 257, 291-347 (2002) · Zbl 1025.20002
[13] Guralnick, R.; Penttila, T.; Praeger, C. E.; Saxl, J., Linear groups with orders having certain large prime divisors, Proc. Lond. Math. Soc., 78, 167-214 (1999) · Zbl 1041.20035
[14] Guralnick, R.; Tiep, P. H., Low-dimensional representations of special linear groups in cross characteristics, Proc. Lond. Math. Soc., 78, 116-138 (1999) · Zbl 0974.20014
[15] Hiss, G.; Malle, G., Low dimensional representations of special unitary groups, J. Algebra, 236, 745-767 (2001) · Zbl 0972.20027
[16] Jansen, C.; Lux, K.; Parker, R. A.; Wilson, R. A., An ATLAS of Brauer Characters (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0831.20001
[17] Kleidman, P. B.; Liebeck, M. W., The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser., vol. 129 (1990), Cambridge Univ. Press · Zbl 0697.20004
[18] Lawther, R.; Liebeck, M. W.; Seitz, G. M., Fixed point ratios in actions of finite exceptional groups of lie type, Pacific J. Math., 205, 393-464 (2002) · Zbl 1058.20001
[19] Liebeck, M. W., On the orders of maximal subgroups of the finite classical groups, Proc. Lond. Math. Soc., 50, 426-446 (1985) · Zbl 0591.20021
[20] Liebeck, M. W.; Seitz, G. M., Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, Math. Surveys Monogr., vol. 180 (2012), Am. Math. Soc. · Zbl 1251.20001
[21] Liebeck, M. W.; Shalev, A., Classical groups, probabilistic methods and the \((2, 3)\)-generation problem, Ann. of Math., 144, 77-125 (1996) · Zbl 0865.20020
[22] Lübeck, F.; Malle, G., \((2, 3)\)-Generation of exceptional groups, J. Lond. Math. Soc., 59, 101-122 (1999) · Zbl 0935.20021
[23] Macbeath, A. M., Generators of the linear fractional group, Proc. Sympos. Pure Math., 12, 14-32 (1969) · Zbl 0192.35703
[24] Malle, G.; Saxl, J.; Weigel, T., Generation of classical groups, Geom. Dedicata, 49, 85-116 (1994) · Zbl 0832.20029
[25] Malle, G.; Testerman, D., Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Stud. Adv. Math., vol. 133 (2011), Cambridge Univ. Press · Zbl 1256.20045
[26] Miller, G. A., On the groups generated by two operators, Bull. Amer. Math. Soc., 7, 424-426 (1901) · JFM 32.0145.03
[27] Miller, G. A., Possible orders of two generators of the alternating and of the symmetric group, Trans. Amer. Math. Soc., 30, 24-32 (1928) · JFM 54.0145.06
[28] Pellegrini, M. A., The \((2, 3)\)-generation of the classical simple groups of dimension 6 and 7, Bull. Aust. Math. Soc., 93, 61-72 (2016) · Zbl 1341.20044
[29] Pellegrini, M. A., The \((2, 3)\)-generation of the special linear groups over finite fields, Bull. Aust. Math. Soc., 95, 48-53 (2017) · Zbl 1369.20046
[30] Pellegrini, M. A.; Prandelli, M.; Tamburini Bellani, M. C., The \((2, 3)\)-generation of the special unitary groups of dimension 6, J. Algebra Appl., 15, Article 1650171 pp. (2016) · Zbl 1351.20028
[31] Pellegrini, M. A.; Tamburini Bellani, M. C., The simple classical groups of dimension less than 6 which are \((2, 3)\)-generated, J. Algebra Appl., 14, Article 1550148 pp. (2015) · Zbl 1325.20030
[32] Pellegrini, M. A.; Tamburini Bellani, M. C.; Vsemirnov, M. A., Uniform \((2, k)\)-generation of the 4-dimensional classical groups, J. Algebra, 369, 322-350 (2012) · Zbl 1272.20055
[33] Sanchini, P.; Tamburini, M. C., Constructive \((2, 3)\)-generation: a permutational approach, Rend. Semin. Mat. Fis. Milano, 64, 141-158 (1994) · Zbl 0860.20039
[34] Shahabi, M. A.; Mohtadifar, H., The characters of finite projective symplectic group \(P S p(4, q)\), (Groups St. Andrews 2001 in Oxford. Vol. II. Groups St. Andrews 2001 in Oxford. Vol. II, London Math. Soc. Lecture Note Ser., vol. 305 (2003), Cambridge Univ. Press), 496-527 · Zbl 1084.20511
[35] Stein, A., \(1 \frac{1}{2} \)-generation of finite simple groups, Beitr. Algebra Geom., 39, 349-358 (1998) · Zbl 0924.20027
[36] Steinberg, R., Generators for simple groups, Canad. J. Math., 14, 277-283 (1962) · Zbl 0103.26204
[37] Suzuki, M., On a class of doubly transitive groups, Ann. of Math., 75, 105-145 (1962) · Zbl 0106.24702
[38] Tabakov, K., \((2, 3)\)-generation of the groups \(P S L_7(q)\), (Proc. of the Forty Second Spring Conf. of the Union of Bulg. Math.. Proc. of the Forty Second Spring Conf. of the Union of Bulg. Math., Borovetz, April 2-6 (2013)), 260-264
[39] Tabakov, K.; Tchakerian, K., \((2, 3)\)-generation of the groups \(P S L_6(q)\), Serdica Math. J., 37, 365-370 (2011) · Zbl 1249.20041
[40] Vsemirnov, M. A., More classical groups which are not \((2, 3)\)-generated, Arch. Math., 96, 123-129 (2011) · Zbl 1220.20025
[41] Wilson, R. A., The Monster is a Hurwitz group, J. Group Theory, 4, 367-374 (2001) · Zbl 0991.20016
[42] Woldar, A. J., On Hurwitz generation and genus actions of sporadic groups, Illinois J. Math., 33, 416-437 (1989) · Zbl 0654.20014
[43] Zsigmondy, K., Zur Theorie der Potenzreste, Monatshefte Math. Phys., 3, 265-284 (1892) · JFM 24.0176.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.