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Simple Groups, Generation and Probabilistic Methods

Published online by Cambridge University Press:  15 April 2019

C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
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
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Summary

It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, (a, b)-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.

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Publisher: Cambridge University Press
Print publication year: 2019

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References

Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469–514.CrossRefGoogle Scholar
Aschbacher, M., On intervals in subgroup lattices of finite groups, J. Amer. Math. Soc. 21 (2008), 809–830.CrossRefGoogle Scholar
Aschbacher, M. and Guralnick, R., Some applications of the first cohomology group, J. Algebra 90 (1984), 446–460.CrossRefGoogle Scholar
Binder, G., The two-element bases of the symmetric group, Izv. Vyss. Ucebn. Zaved. Matematika 90 (1970), 9–11.Google Scholar
Bradley, J.D. and Holmes, P.E., Improved bounds for the spread of sporadic groups, LMS J. Comput. Math. 10 (2007), 132–140.CrossRefGoogle Scholar
Bradley, J.D. and Moori, J., On the exact spread of sporadic simple groups, Comm. Algebra 35 (2007), 2588–2599.CrossRefGoogle Scholar
Bray, J.N., Holt, D.F. and Roney-Dougal, C.M., The Maximal Subgroups of the Lowdimensional Finite Classical Groups, London Math. Soc. Lecture Note Series, vol. 407, Cambridge University Press, 2013.CrossRefGoogle Scholar
Brenner, J.L. and Wiegold, J., Two-generator groups I, Michigan Math. J. 22 (1975), 53–64.Google Scholar
Breuer, T., GAP computations concerning Hamiltonian cycles in the generating graphs of finite groups, preprint, 2012 (arxiv:0911.5589).Google Scholar
Breuer, T., Guralnick, R.M. and Kantor, W.M., Probabilistic generation of finite simple groups, II, J. Algebra 320 (2008), 443–494.Google Scholar
Breuer, T., Guralnick, R.M., Lucchini, A., Maróti, A., and Nagy, G.P., Hamiltonian cycles in the generating graph of finite groups, Bull. London Math. Soc. 42 (2010), 621–633.CrossRefGoogle Scholar
Burness, T.C., Fixed point ratios in actions of finite classical groups, I, J. Algebra 309 (2007), 69–79.Google Scholar
Burness, T.C. and Guest, S. On the uniform spread of almost simple linear groups, Nagoya Math. J. 209 (2013), 35–109.CrossRefGoogle Scholar
Burness, T.C., Liebeck, M.W. and Shalev, A., Generation and random generation: from simple groups to maximal subgroups, Adv. Math. 248 (2013), 59–95.CrossRefGoogle Scholar
Burness, T.C., Liebeck, M.W. and Shalev, A., Generation of second maximal subgroups and the existence of special primes, Forum Math. Sigma 5 (2017), e25, 41 pp.CrossRefGoogle Scholar
Cameron, P.J., Lucchini, A. and Roney-Dougal, C.M., Generating sets of finite groups, Trans. Amer. Math. Soc. 370 (2018), 6751–6770.CrossRefGoogle Scholar
Conder, M.D.E., Generators for alternating and symmetric groups, J. London Math. Soc. 22 (1980), 75–86.Google Scholar
Conder, M.D.E., An update on Hurwitz groups, Groups Complex. Cryptol. 2 (2010), 35–49.CrossRefGoogle Scholar
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A., Atlas of Finite Groups, Oxford University Press, 1985.Google Scholar
Dalla Volta, F. and Lucchini, A., Generation of almost simple groups, J. Algebra 178 (1995), 194–223.Google Scholar
Dixon, J.D., The probability of generating the symmetric group, Math. Z. 110 (1969), 199-205.CrossRefGoogle Scholar
Dixon, J.D. and Mortimer, B., Permutation groups, Graduate Texts in Math., vol. 163, Springer-Verlag, New York, 1996.CrossRefGoogle Scholar
Erdem, F., On the generating graphs of symmetric groups, J. Group Theory 21 (2018), 629–649.CrossRefGoogle Scholar
Erdős, P. and Turán, P., On some problems of a statistical group-theory, II, Acta. Math. Acad. Sci. Hung. 18 (1967), 151–163.Google Scholar
Everitt, B., Alternating quotients of Fuchsian groups, J. Algebra 223 (2000), 457–476.CrossRefGoogle Scholar
Fairbairn, B., The exact spread of M23 is 8064, Int. J. Group Theory 1 (2012), 1–2.Google Scholar
Fairbairn, B., New upper bounds on the spreads of the sporadic simple groups, Comm. Algebra 40 (2012), 1872–1877.CrossRefGoogle Scholar
Guest, S., Morris, J., Praeger, C.E. and Spiga, P., On the maximum orders of elements of finite almost simple groups and primitive permutation groups, Trans. Amer. Math. Soc. 367 (2015), 7665–7694.CrossRefGoogle Scholar
Guralnick, R.M. and Kantor, W.M., Probabilistic generation of finite simple groups, J. Algebra 234 (2000), 743–792.CrossRefGoogle Scholar
Guralnick, R., Pentilla, T., Praeger, C.E., and Saxl, J., Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. 78 (1999), 167–214.CrossRefGoogle Scholar
Guralnick, R.M. and Shalev, A., On the spread of finite simple groups, Combinatorica 23 (2003), 73–87.CrossRefGoogle Scholar
Harper, S., On the uniform spread of almost simple symplectic and orthogonal groups, J. Algebra 490 (2017), 330–371.CrossRefGoogle Scholar
Jaikin-Zapirain, A. and Pyber, L., Random generation of finite and profinite groups and group enumeration, Annals of Math. 173 (2011), 769–814.CrossRefGoogle Scholar
Jambor, S., Litterick, A. and Marion, C., On finite simple images of triangle groups, Israel J. Math. 227 (2018), 131–162.CrossRefGoogle Scholar
Kantor, W.M. and Lubotzky, A., The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67–87.CrossRefGoogle Scholar
King, C.S.H., Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017), 153–173.CrossRefGoogle Scholar
Kleidman, P.B. and Liebeck, M.W., The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Series, vol. 129, Cambridge University Press, 1990.CrossRefGoogle Scholar
Larsen, M., Lubotzky, A. and Marion, C., Deformation theory and finite simple quotients of triangle groups I, J. Eur. Math. Soc. (JEMS) 16 (2014), 1349–1375.CrossRefGoogle Scholar
Lawther, R., Liebeck, M.W. and Seitz, G.M., Fixed point ratios in actions of finite exceptional groups of Lie type, Pacific J. Math. 205 (2002), 393–464.Google Scholar
Liebeck, M.W., Probabilistic and asymptotic aspects of finite simple groups, in Probabilistic group theory, combinatorics, and computing, 1–34, Lecture Notes in Math., 2070, Springer, London, 2013.CrossRefGoogle Scholar
Liebeck, M.W. and Saxl, J., Minimal degrees of primitive permutation groups, with an application to monodromy groups of coverings of Riemann surfaces, Proc. London Math. Soc. 63 (1991), 266–314.Google Scholar
Liebeck, M.W. and Seitz, G.M., A survey of of maximal subgroups of exceptional groups of Lie type, in Groups, combinatorics & geometry (Durham, 2001), 139–146, World Sci. Publ., River Edge, NJ, 2003.CrossRefGoogle Scholar
Liebeck, M.W. and Shalev, A., The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103–113.CrossRefGoogle Scholar
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
Liebeck, M.W. and Shalev, A., Classical groups, probabilistic methods, and the (2, 3)-generation problem, Annals of Math. 144 (1996), 77–125.CrossRefGoogle Scholar
Liebeck, M.W. and Shalev, A., Random (r, s)-generation of finite classical groups, Bull. London Math. Soc. 34 (2002), 185–188.CrossRefGoogle Scholar
Liebeck, M.W. and Shalev, A., Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks, J. Algebra 276 (2004), 552–601.CrossRefGoogle Scholar
Liebeck, M.W. and Shalev, A., Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), 317–367.CrossRefGoogle Scholar
Lübeck, F. and Malle, G., (2, 3)-generation of exceptional groups, J. London Math. Soc. 59 (1999), 109–122.CrossRefGoogle Scholar
Lubotzky, A., The expected number of random elements to generate a finite group, J. Algebra 257 (2002), 452–459.CrossRefGoogle Scholar
Lucchini, A. and Menegazzo, F., Generators for finite groups with a unique minimal normal subgroup, Rend. Semin. Mat. Univ. Padova 98 (1997), 173–191.Google Scholar
Macbeath, A.M., Generators of the linear fractional groups, Proc. Sympos. Pure Math., vol. XII (Amer. Math. Soc., 1969), pp.14–32.CrossRefGoogle Scholar
Malle, G., Saxl, J. and Weigel, T., Generation of classical groups, Geom. Dedicata 49 (1994), 85–116.CrossRefGoogle Scholar
Mann, A. and Shalev, A., Simple groups, maximal subgroups, and probabilistic aspects of profinite groups, Israel J. Math. 96 (1996), 449–468.CrossRefGoogle Scholar
Marion, C., Triangle groups and PSL2(q), J. Group Theory 12 (2009), 689–708.CrossRefGoogle Scholar
Marion, C., On triangle generation of finite groups of Lie type, J. Group Theory 13 (2010), 619–648.CrossRefGoogle Scholar
McIver, A. and Neumann, P., Enumerating finite groups, Quart. J. Math. Oxford 38 (1987), 473–488.CrossRefGoogle Scholar
Menezes, N.E., Quick, M. and Roney-Dougal, C.M., The probability of generating a finite simple group, Israel J. Math. 198 (2013), 371–392.CrossRefGoogle Scholar
Miller, G.A., On the groups generated by two operators, Bull. Amer. Math. Soc. 7 (1901), 424–426.CrossRefGoogle Scholar
Morgan, L. and Roney-Dougal, C.M., A note on the probability of generating alternating or symmetric groups, Arch. Math. (Basel) 105 (2015), 201–204.CrossRefGoogle Scholar
Netto, E., Substitutionentheorie und ihre Anwendungen auf die Algebra, Teubner, Leipzig, 1882; English transl. 1892, second edition, Chelsea, New York, 1964.Google Scholar
Pak, I., On probability of generating a finite group, preprint, 1999.Google Scholar
Pálfy, P.P., On Feit’s examples of intervals in subgroup lattices, J. Algebra 116 (1988), 471–479.CrossRefGoogle Scholar
Pellegrini, M.A., The (2, 3)-generation of the special linear groups over finite fields, Bull. Aust. Math. Soc. 95 (2017), 48–53.CrossRefGoogle Scholar
Pellegrini, M.A. and Tamburini, M.C., Finite simple groups of low rank: Hurwitz generation and (2, 3)-generation, Int. J. Group Theory 4 (2015), 13–19.Google Scholar
Piccard, S., Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116 (1939), 752–767.CrossRefGoogle Scholar
Scott, L.L., Matrices and cohomology, Annals of Math. 105 (1977), 473–492.CrossRefGoogle Scholar
Shalev, A., Probabilistic group theory and Fuchsian groups, in Infinite groups: geometric, combinatorial and dynamical aspects, 363–388, Progr. Math., 248, Birkhäuser, Basel, 2005. 2CrossRefGoogle Scholar
Stein, A., 1 ½-generation of finite simple groups, Beiträge Algebra Geom. 39 (1998), 349–358.Google Scholar
Steinberg, R., Generators for simple groups, Canad. J. Math. 14 (1962), 277–283.CrossRefGoogle Scholar
Suzuki, M., On a class of doubly transitive groups, Annals of Math. 75 (1962), 105–145.CrossRefGoogle Scholar
Weigel, T.S., Generation of exceptional groups of Lie-type, Geom. Dedicata 41 (1992), 63–87.CrossRefGoogle Scholar
Wielandt, H., Finite Permutation Groups, Academic Press, New York (1964).Google Scholar
Woldar, A.J., On Hurwitz generation and genus actions of sporadic groups, Illinois Math. J. 33 (1989), 416–437.CrossRefGoogle Scholar
Zsigmondy, K., Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), 265–284.Google Scholar

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