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Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective

Published online by Cambridge University Press:  15 April 2019

By
Harald A. Helfgott
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
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

By now, we have a product theorem in every finite simple group G of Lie type, with the strength of the bound depending only in the rank of G. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Altn, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Altn, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Altn – not of full strength, as that would be impossible, but strong enough to imply the diameter bound.

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Chapter
Information
Groups St Andrews 2017 in Birmingham , pp. 300 - 345
Publisher: Cambridge University Press
Print publication year: 2019

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References

Babai, L., On the order of doubly transitive permutation groups. Invent. math. 65(3) 1981/82, 473–484.Google Scholar
Babai, L., On the length of subgroup chains in the symmetric group. Comm. Algebra 14(9) 1986, 1729–1736.CrossRefGoogle Scholar
Babai, L., On the diameter of Eulerian orientations of graphs. In: Proc. 17th ACMSIAM Symp. on Discrete Algorithms. ACM, New York, 2006, 822–831Google Scholar
Babai, L., Beals, R., and Seress, Á., On the diameter of the symmetric group: polynomial bounds. In: Proc. 15th ACM-SIAM Symp. on Discrete Algorithms. ACM, New York, 2004, 1108–1112.Google Scholar
Babai, L., Luks, E. M., and Seress, Á., Fast management of permutation groups. I. SIAM J. Comput. 26(5) 1997, 1310–1342.CrossRefGoogle Scholar
Babai, L., Nikolov, N., and Pyber, L., Product growth and mixing in finite groups. In: Proc. 19th ACM-SIAM Symp. on Discrete Algorithms, ACM, New York, 2008, 248–257.Google Scholar
Babai, L. and Seress, Á., On the degree of transitivity of permutation groups: a short proof. J. Combin. Theory Ser. A 45(2) 1987, 310–315.CrossRefGoogle Scholar
Babai, L. and Seress, Á., On the diameter of permutation groups. Eur. J. Comb. 13(4) 1992, 231–243.CrossRefGoogle Scholar
Bourgain, J. and Gamburd, A., Uniform expansion bounds for Cayley graphs of SL2(Fp). Ann. of Math. (2) 167(2) 2008, 625–642.CrossRefGoogle Scholar
Bourgain, J., Gamburd, A., and Sarnak, P., Affine linear sieve, expanders, and sumproduct. Invent. math. 179(3), 2010, 559–644.CrossRefGoogle Scholar
Bourgain, J., Gamburd, A., and Sarnak, P., Generalization of Selberg’s 3 theorem and affine sieve. Acta Math. 207(2) 2011, 255–290.CrossRefGoogle Scholar
Breuillard, E., Green, B., and Tao, T.. Approximate subgroups of linear groups. Geom. Funct. Anal. 21(4), 2011, 774–819.CrossRefGoogle Scholar
Cameron, P. J., Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13(1), 1981, 1–22.CrossRefGoogle Scholar
Dixon, J. D. and Mortimer, B.. Permutation groups, Volume 163 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996.CrossRefGoogle Scholar
Eskin, A., Mozes, Sh, and Oh, H., On uniform exponential growth for linear groups. Invent. math. 160(1) 2005, 1–30.CrossRefGoogle Scholar
Glasby, S. P., Praeger, C. E., Rosa, K., and Verret, Gabriel, Bounding the composition length of primitive permutation groups and completely reducible linear groups. Preprint. Available as https://arxiv.org/abs/1712.05520.Google Scholar
Gowers, W. T., Quasirandom groups. Combin. Probab. Comput. 17(3), 2008, 363–387.CrossRefGoogle Scholar
Helfgott, H. A., Growth and generation in SL2(Z/pZ). Ann. of Math. (2) 167(2), 2008, 601–623.CrossRefGoogle Scholar
Helfgott, H. A., Growth in SL3(Z/pZ). J. Eur. Math. Soc. 13(3) 2011, 761–851.Google Scholar
Helfgott, H. A., Growth in groups: ideas and perspectives. Bull. Amer. Math. Soc. 52(3) 2015, 357–413.CrossRefGoogle Scholar
Helfgott, H. A.. When is the union of a graph and a random permutation thereof connected? MathOverflow. https://mathoverflow.net/q/286057 (2017-11-16).Google Scholar
Helfgott, H. A. and Seress, Á., On the diameter of permutation groups. Ann. of Math. (2) 179(2), 2014, 611–658.CrossRefGoogle Scholar
Kappe, L.-C. and Morse, R. F., On commutators in groups. In: Groups St. Andrews 2005. Vol. 2, volume 340 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, 2007, 531–558.Google Scholar
Liebeck, M. W., On graphs whose full automorphism group is an alternating group or a finite classical group. Proc. London Math. Soc. (3) 47(2) 1983, 337–362.Google Scholar
Liebeck, M. W., On minimal degrees and base sizes of primitive permutation groups. Arch. Math. (Basel) 43(1) 1984, 11–15.CrossRefGoogle Scholar
Liebeck, M. W. and Shalev, A., Diameters of finite simple groups: sharp bounds and applications. Ann. of Math. (2) 154(2) 2001, 383–406.CrossRefGoogle Scholar
Luks, E. M. and McKenzie, P., Parallel algorithms for solvable permutation groups. J. Comput. System Sci. 37(1) 1988, 39–62.CrossRefGoogle Scholar
Maróti, A., On the orders of primitive groups. J. Algebra 258(2) 2002, 631–640.CrossRefGoogle Scholar
Miller, G. A., On the commutators of a given group. Bulletin of the American Mathematical Society 6(3) 1899, 105–109.CrossRefGoogle Scholar
Praeger, C. E., Pyber, L., Spiga, P., and Szabó, E., Graphs with automorphism groups admitting composition factors of bounded rank. Proc. Amer. Math. Soc. 140(7), 2012, 2307–2318.Google Scholar
Pyber, L., On the orders of doubly transitive permutation groups, elementary estimates. J. Combin. Theory Ser. A 62(2) 1993, 361–366.CrossRefGoogle Scholar
Pyber, L. and Szabó, E., Growth in finite simple groups of Lie type. J. Amer. Math. Soc. 29(1), 2016, 95–146.Google Scholar
Seress, Á., Permutation group algorithms. Cambridge University Press, 2003.CrossRefGoogle Scholar
Sims, C. C., Computational methods in the study of permutation groups. In: Computational Problems in Abstract Algebra, Pergamon, Oxford, 1970, 169–183Google Scholar
Spiga, P., Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups. J. Group Theory 15(1), 2012, 23–35.CrossRefGoogle Scholar

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