Finite simple groups of low rank: Hurwitz generation and (2, 3)-generation

Document Type : Ischia Group Theory 2014

Authors

Universita Cattolica del Sacro Cuore

Abstract

‎Let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $\le 7$ over‎ ‎an algebraically closed field $F$ of characteristic $p\geq 0$‎. ‎We survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

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J. Cohen (1981). On non-Hurwitz groups and non-congruence subgroups of the modular group. Glasgow Math. J.. 22, 1-7 M. Conder (2010). An update on Hurwitz groups. Groups Complex. Cryptol.. 2, 35-49 L. Di Martino and M. C. Tamburini (1991). 2 -Generation of finite simple groups and some related topics. Generators and relations in groups and geometries, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht. 333, 195-233 L. Finkelstein and A. Rudvalis (1973). Maximal subgroups of the Hall-Janko-Wales group. J. Algebra. 24, 486-493 D. Garbe (1978). Ubereine classe von arithmetisch definierbaren Normalteilern der Modulgruppe. Math. Ann.. 235, 195-215 G. Hiss and G. Malle (2001). Corrigenda: low-dimensional representations of quasi-simple groups. LMS LMS J. Comput. Math.. 4, 22-63 A. Hurwitz (1892). Ueb er algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann.. 41, 403-442 P. Kleidman and M. Liebeck (1990). The subgroup structure of the finite classical groups. London Mathematical So ciety Lecture Note Series, Cambridge University Press, Cambridge. 129 F. Klein (1879). Uber die Transformation sieb enter Ordnung der elliptischen Functionen. Math. Ann.. 14, 428-471 A. S. Kondratiev (1998). Finite linear groups of small degree. London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge. 249, 139-148 M. W. Liebeck and A. Shalev (1996). Classical groups, probabilistic methods, and the (2 ; 3)-generation problem. Ann. of Math. (2). 144, 77-125 A. M. Macbeath (1969). Generators for the linear fractional groups. Proc. Simp. Pure Math.. 12, 14-32 G. Malle (1990). Hurwitz groups and G_2 (q). Canad. Math. Bull.. 33, 349-357 C. Marion (2010). On triangle generation of finite groups of Lie type. J. Group Theory. 13, 619-648 G. A. Miller (1901). On the groups generated by two operators. Bul l. Amer. Math. Soc.. 7, 424-426 M. A. Pellegrini The (2 ; 3) -generation of the classical simple groups of dimension 6 and 7. to app ear in Bull. Austr. Math. So c., DOI: 10.1017 /S0004972715000763, http://arxiv.org/abs/1503.07056. M. A. Pellegrini, M. Prandelli and M. C. Tamburini Bellani The (2 ; 3) -generation of the special unitary groups of dimension 6. http://arxiv.org/abs/1409.3411. M. A. Pellegrini and M. C. Tamburini Bellani (2015). Scott's formula and Hurwitz groups. J. Algebra. 443, 126-141 M. A. Pellegrini and M. C. Tamburini Bellani (2015). The simple classical groups of dimension less than 6 which are (2 ; 3)-generated. J. Algebra Appl., DOI: 10.1142/S0219498815501480. 14, 15