Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
Published online by Cambridge University Press: 15 April 2019
Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
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.
- Type
- Chapter
- Information
- Groups St Andrews 2017 in Birmingham , pp. 300 - 345Publisher: Cambridge University PressPrint publication year: 2019
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 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
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
- 3
- Cited by