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3 - The spread of finite and infinite groups

Published online by Cambridge University Press:  21 November 2024

By
Scott Harper
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal  and
D. I. Stewart
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
D. I. Stewart
Affiliation:
University of Manchester
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Summary

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is contained in a generating pair. More recently, this result has been generalised in three different directions, which form the basis of this survey article. First, we look at some stronger forms of $\frac{3}{2}$-generation that the finite simple groups satisfy, which are described in terms of spread and uniform domination. Next, we discuss the recent classification of the finite $\frac{3}{2}$-generated groups. Finally, we turn our attention to infinite groups, and we focus on the recent discovery that the finitely presented simple groups of Thompson are also $\frac{3}{2}$-generated, as are many of their generalisations. Throughout the article we pose open questions in this area, and we highlight connections with other areas of group theory.

Keywords

group generationspreadsimple groupThompsons groups
Type
Chapter
Information
Groups St Andrews 2022 in Newcastle , pp. 74 - 117
Publisher: Cambridge University Press
Print publication year: 2024

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References

[1] Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514.CrossRefGoogle Scholar
[2] Aschbacher, M. and Guralnick, R., Some applications of the first cohomology group, J. Algebra 90 (1984), 446–460.CrossRefGoogle Scholar
[3] Binder, G. J., The bases of the symmetric group, Izv. Vyssh. Uchebn. Zaved. Mat. 78 (1968), 19–25.Google Scholar
[4] Binder, G. J., Certain complete sets of complementary elements of the symmetric and the alternating group of the nth degree, Mat. Zametiki 7 (1970), 173–180.Google Scholar
[5] Binder, G. J., The two-element bases of the symmetric group, Izv. Vyssh. Uchebn. Zaved. Mat. 90 (1970), 9–11.Google Scholar
[6] Binder, G. J., The inclusion of the elements of an alternating group of even degree in a two-element basis, Izv. Vyssh. Uchebn. Zaved. Mat. 135 (1973), 15–18.Google Scholar
[7] Blackburn, S., Sets of permutations that generate the symmetric group pairwise, J. Combin. Theory Ser. A 113 (2006), 1572–1581.Google Scholar
[8] Bleak, C., Donoven, C., Harper, S. and Hyde, J., Generating simple vigorous groups, in preparation.Google Scholar
[9] Bleak, C., Elliott, L. and Hyde, J., Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated, J. Inst. Math. Jussieu, to appear. 2Google Scholar
[10] Bleak, C., S. Harper and Skipper, R., Thompson’s group T is 3-generated, Math, Israel J.., to appear.Google Scholar
[11] Bleak, C. and Lanoue, D., A family of non-isomorphic results, Geom. Dedicata 146 (2010), 21–26.Google Scholar
[12] Bleak, C. and Quick, M., The infinite simple group V of Richard J. Thompson: presentations by permutations, Groups Geom. Dyn. 11 (2017), 1401–1436.Google Scholar
[13] Bois, J.-M., Generators of simple Lie algebras in arbitrary characteristics, Math. Z. 262 (2009), 715–741.Google Scholar
[14] Bradley, J. D. and Holmes, P. E., Improved bounds for the spread of sporadic groups, LMS J. Comput. Math. 10 (2007), 132–140.Google Scholar
[15] Brenner, J. L. and Wiegold, J., Two generator groups, I, Michigan Math. J. 22 (1975), 53–64.Google Scholar
[16] Breuer, T., Guralnick, R. M. and Kantor, W. M., Probabilistic generation of finite simple groups, II, J. Algebra 320 (2008), 443–494.CrossRefGoogle Scholar
[17] Breuer, T., Guralnick, R. M., Lucchini, A., Maróti, A. and Nagy, G. P., Hamiltonian cycles in the generating graphs of finite groups, Bull. Lond. Math. Soc. 42 (2010), 621–633.Google Scholar
[18] Brin, M. G., Higher dimensional Thompson groups, Geom. Dedicata 108 (2004), 163–192.Google Scholar
[19] Brin, M. G., On the baker’s maps and the simplicity of the higher dimensional Thompson’s groups nV, Publ. Mat. 54 (2010), 433–439.Google Scholar
[20] Britnell, J. R., Evseev, A., Guralnick, R. M., Holmes, P. E. and Maróti, A., Sets of elements that pairwise generate a linear group, J. Combin. Theory Ser. A 115 (2008), 442–465.Google Scholar
[21] Burger, M. and Mozes, S., Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 151–194.Google Scholar
[22] Burness, T. C., Fixed point ratios in actions of finite classical groups, I, J. Algebra 309 (2007), 69–79.CrossRefGoogle Scholar
[23] Burness, T. C., Fixed point ratios in actions of finite classical groups, II, J. Algebra 309 (2007), 80–138.CrossRefGoogle Scholar
[24] Burness, T. C., Fixed point ratios in actions of finite classical groups, III, J. Algebra 314 (2007), 693–748.CrossRefGoogle Scholar
[25] Burness, T. C., Fixed point ratios in actions of finite classical groups, IV, J. Algebra 314 (2007), 749–788.CrossRefGoogle Scholar
[26] Burness, T. C., Simple groups, fixed point ratios and applications, in Local Representation Theory and Simple Groups, EMS Series of Lectures in Mathematics, European Mathematical Society, 2018, 267–322.CrossRefGoogle Scholar
[27] Burness, T. C., Simple groups, generation and probabilistic methods, in Proceedings of Groups St Andrews 2017, London Math. Soc. Lecture Note Series, vol. 455, Cambridge University Press, 2019, 200–229.Google Scholar
[28] Burness, T. C. and Guest, S., On the uniform spread of almost simple linear groups, Nagoya Math. J. 209 (2013), 35–109.Google Scholar
[29] Burness, T. C., Guralnick, R. M. and Harper, S., The spread of a finite group, Ann. of Math. 193 (2021), 619–687.CrossRefGoogle Scholar
[30] Burness, T. C. and Harper, S., On the uniform domination number of a finite simple group, Trans. Amer. Math. Soc. 372 (2019), 545–583.Google Scholar
[31] Burness, T. C. and Harper, S., Finite groups, 2-generation and the uniform domination number, Israel J. Math. 239 (2020), 271–367.Google Scholar
[32] Burness, T. C., M. .W. Liebeck and Shalev, A., Base sizes for simple groups and a conjecture of Cameron, Proc. Lond. Math. Soc. 98 (2009), 116–162.Google Scholar
[33] Burness, T. C. and Thomas, A. R., Normalisers of maximal tori and a conjecture of Vdovin, J. Algebra 619 (2023), 459–504.CrossRefGoogle Scholar
[34] Cameron, P. J., Graphs defined on groups, Int. J. Group Theory 11 (2022), 53–107.Google Scholar
[35] Cannon, J. W., Floyd, W. J. and Parry, W. R., Introductory notes on Richard Thompson’s groups, Enseign. Math. 42 (1996), 1–44.Google Scholar
[36] Celler, F., Leedham-Green, C. R., Murray, S. H., Niemeyer, A. C. and O’Brien, E. A., Generating random elements of a finite group, Comm. Algebra 23 (1995), 4391–4948.Google Scholar
[37] Cleary, S., Thompson’s group, in Office Hours with a Geometric Group Theorist, Princeton University Press, 2017, 331–357.CrossRefGoogle Scholar
[38] Cox, C. G., On the spread of infinite groups, Proc. Edinb. Math. Soc. 65 (2022), 214–228.Google Scholar
[39] Crestani, E. and Lucchini, A., The generating graph of finite soluble groups, Israel J. of Math. 198 (2013), 63–74.Google Scholar
[40] Crestani, E. and Lucchini, A., The non-isolated vertices in the generating graph of direct powers of simple groups, J. Algebraic Combin. 37 (2013), 249–263.Google Scholar
[41] Deshpande, T., Shintani descent for algebraic groups and almost simple characters of unipotent groups, Compos. Math. 152 (2016), 1697–1724.CrossRefGoogle Scholar
[42] Donoven, C. and Harper, S., Infinite 3/2-generated groups, Bull. Lond. Math. Soc. 52 (2020), 657–673.Google Scholar
[43] Duyan, H., Halasi, Z. and Maróti, A., A proof of Pyber’s base size conjecture, Adv. Math. 331 (2018), 720–747.CrossRefGoogle Scholar
[44] Dydak, J., 1-movable continua need not be pointed 1-movable, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 559–562.Google Scholar
[45] Dye, R. H., Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979), 202–221.CrossRefGoogle Scholar
[46] Evans, M. J., T-systems of certain finite simple groups, Math. Proc. Camb. Phil. Soc. 113 (1993), 9–22.Google Scholar
[47] Fairbairn, B., The exact spread of M23 is 8064, Int. J. Group Theory 1 (2012), 1–2.Google Scholar
[48] Fairbairn, B., New upper bounds on the spreads of sporadic simple groups, Comm. Algebra 40 (2012), 1872–1877.CrossRefGoogle Scholar
[49] Flavell, P., Finite groups in which every two elements generate a soluble subgroup, Invent. Math. 121 (1995), 279–285.CrossRefGoogle Scholar
[50] Flavell, P., Generation theorems for finite groups, in Groups and combinatorics – in memory of Michio Suzuki, Adv. Stud. Pure. Math., vol. 32, Math. Soc. Japan, 2001, 291–300.Google Scholar
[51] Fulman, J. and Guralnick, R. M., Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012), 3023–3070.Google Scholar
[52] Ghys, E. and Sergiescu, V., Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987), 185–239.Google Scholar
[53] Golan Polak, G., The generation problem in Thompson group F, Mem. Amer. Math. Soc. 292 (2023), v+94.Google Scholar
[54] Golan Polak, G., On maximal subgroups of Thompson’s group F, preprint, arxiv:2209.03244.Google Scholar
[55] Polak, G. Golan, Thompson’s group F is almost 3 -generated, Bull. Lond. Math. Soc. 55 (2023), 2144–2157.Google Scholar
[56] Goldstein, D. and Guralnick, R. M., Generation of Jordan algebras and symmetric matrices, in preparation.Google Scholar
[57] Guba, V. S., A finite generated simple group with free 2-generated subgroups, Sibirsk. Mat. Zh. 27 (1986), 50–67.Google Scholar
[58] Guba, V. and Sapir, M., Diagram groups, Mem. Amer. Math. Soc. 130 (1997), viii+117.Google Scholar
[59] Guralnick, R. M. and Kantor, W. M., Probabilistic generation of finite simple groups, J. Algebra 234 (2000), 743–792.CrossRefGoogle Scholar
[60] Guralnick, R., B. Kunyavski˘ı, E. Plotkin and Shalev, A., Thompson-like characterizations of the solvable radical, J. Algebra 300 (2006), 363–375.Google Scholar
[61] Guralnick, R. M. and Malle, G., Simple groups admit Beauville structures, Lond, J.. Math. Soc. 85 (2012), 694–721.Google Scholar
[62] Guralnick, R. M., Penttila, T., Praeger, C. E. and Saxl, J., Linear groups with orders having certain large prime divisors, Proc. Lond. Math. Soc. 78 (1997), 167–214.Google Scholar
[63] Guralnick, R., Plotkin, E. and Shalev, A., Burnside-type problems related to solvability, Internat. J. Algebra Comput. 17 (2007), 1033–1048.Google Scholar
[64] Guralnick, R. M. and Saxl, J., Generation of finite almost simple groups by conjugates, J. Algebra 268 (2003), 519–571.CrossRefGoogle Scholar
[65] Guralnick, R. M. and Shalev, A., On the spread of finite simple groups, Combinatorica 23 (2003), 73–87.CrossRefGoogle Scholar
[66] Halasi, Z., On the base size of the symmetric group acting on subsets, Stud. Sci. Math. Hung. 49 (2012), 492–500.Google Scholar
[67] Hall, P., The Eulerian functions of a group, Quart. J. Math. 7 (1936), 134–151.CrossRefGoogle Scholar
[68] Harper, S., On the uniform spread of almost simple symplectic and orthogonal groups, J. Algebra 490 (2017), 330–371.CrossRefGoogle Scholar
[69] Harper, S., Shintani descent, simple groups and spread, J. Algebra 578 (2021), 319–355.CrossRefGoogle Scholar
[70] Harper, S., The spread of almost simple classical groups, Lecture Notes in Mathematics, vol. 2286, Springer, 2021.Google Scholar
[71] Higman, G., Finitely presented infinite simple groups, Notes on Pure Mathematics, vol. 8, Department of Mathematics, I.A.S., Australia National University, Canberra, 1974.Google Scholar
[72] Ionescu, T., On the generators of semisimple Lie algebras, Linear Algebra Appl. 15 (1976), 271–292.CrossRefGoogle Scholar
[73] Kantor, W. M. and Lubotzky, A., The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67–87.Google Scholar
[74] Kantor, W. M., Lubotzky, A. and Shalev, A., Invariable generation and the Chebotarev invariant of a finite group, J. Algebra 348 (2011), 302–314.CrossRefGoogle Scholar
[75] Kawanaka, N., On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan 29 (1977), 425–450.Google Scholar
[76] Khukhro, E. I. and Mazurov, V. D. (editors), The Kourovka Notebook: Unsolved Problems in Group Theory, 20th Edition, Novosibirsk, 2022, arxiv:1401.0300.Google Scholar
[77] Liebeck, M. W., O’Brien, E. A., Shalev, A. and Tiep, P. H., The Ore conjecture, J. Eur. Math. Soc. 12 (2010), 939–1008.Google Scholar
[78] Liebeck, M. W. and Saxl, J., Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. Lond. Math. Soc. 63 (1991), 266–314.Google Scholar
[79] Liebeck, M. W. and Shalev, A., The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103–113.Google Scholar
[80] Liebeck, M. W. and Shalev, A., Probabilistic methods, and the (2, 3)-generation problem, Ann. of Math. 144 (1996), 77–125.CrossRefGoogle Scholar
[81] Liebeck, M. W. and Shalev, A., Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, J. Algebra 184 (1996), 31–57.CrossRefGoogle Scholar
[82] Liebeck, M. W. and Shalev, A., Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), 497–520.Google Scholar
[83] Lübeck, F. and Malle, G., (2,3)-generation of exceptional groups, J. Lond. Math. Soc. 59 (1999), 109–122.Google Scholar
[84] Lubotzky, A., Images of word maps in finite simple groups, Glasg. Math. J. 56 (2014), 465–469.Google Scholar
[85] Lucchini, A. and Maróti, A., On the clique number of the generating graph of a finite group, Proc. Amer. Math. Soc. 137 (2009), 3207–3217.Google Scholar
[86] Lucchini, A. and Maróti, A., Some results and questions related to the generating graph of a finite group, in Ischia Group Theory 2008, World Scientific Publishing, 2009, 183–208.Google Scholar
[87] Mason, D. R., On the 2-generation of certain finitely presented infinite simple groups, J. Lond. Math. Soc. 16 (1977), 229–231.CrossRefGoogle Scholar
[88] Ol’shanskii, A. Y., An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 309–321.Google Scholar
[89] Osin, D. and Thom, A., Normal generation and l2-Betti numbers of groups, Math. Ann. 355 (2013), 1331–1347.CrossRefGoogle Scholar
[90] Pak, I., What do we know about the product replacement algorithm?, in Groups and computation, III (Columbus, OH, 1999), Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, 2001, 301–347.CrossRefGoogle Scholar
[91] Piccard, S., Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116 (1939), 752–767.CrossRefGoogle Scholar
[92] Quick, M., Permutation-based presentations for Brin’s higherdimensional Thompson groups nV, J. Aust. Math. Soc. to appear.Google Scholar
[93] Shintani, T., Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Japan 28 (1976), 396–414.Google Scholar
[94] Stein, A., 1 1 -generation of finite simple groups, Contrib. Algebra and Geometry 39 (1998), 349–358.Google Scholar
[95] Steinberg, R., Generators for simple groups, Canadian J. Math. 14 (1962), 277–283.Google Scholar
[96] Thompson, J. G., Nonsolvable groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437.Google Scholar
[97] Thompson, R. J., widely circulated handwritten notes (1965), 1–11.Google Scholar
[98] Thompson, R. J., Embeddings into finitely generated simple groups which preserve the word problem, in Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), North-Holland, 1980, 401–441.Google Scholar
[99] Weigel, T. S., Generation of exceptional groups of Lie-type, Geom. Dedicata 41 (1992), 63–87.Google Scholar
[100] Wielandt, H., Finite Permutation Groups, Academic Press, 1964.Google Scholar
[101] Woldar, A., The exact spread of the Mathieu group M11, J. Group Theory 10 (2007), 167–171.Google Scholar

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