Fixed point ratios in actions of finite classical groups. III. (English) Zbl 1133.20003
This is the third in a series of four papers on fixed point ratios for actions of classical groups [see T. C. Burness, part I, J. Algebra 309, No. 1, 69-79 (2007; Zbl 1128.20003), for more details on the set-up].
The main theorem of these papers is the following: if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then either \(\text{fpr}(x)\lesssim|x^G|^{-\frac 12}\) for all \(x\in G\) of prime order, or the pair \((G,\Omega)\) is one of a small number of known exceptions.
The proof of this result first reduces to primitive actions and then considers the different possibilities for the stabiliser \(G_\omega\) of a point \(\omega\in\Omega\); in the paper under review the result is proved when \(G_\omega\) lies in a maximal subgroup from one of the Aschbacher families \(\mathcal C_2\) or \(\mathcal C_3\) [see M. Aschbacher, Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)].
The main theorem of these papers is the following: if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then either \(\text{fpr}(x)\lesssim|x^G|^{-\frac 12}\) for all \(x\in G\) of prime order, or the pair \((G,\Omega)\) is one of a small number of known exceptions.
The proof of this result first reduces to primitive actions and then considers the different possibilities for the stabiliser \(G_\omega\) of a point \(\omega\in\Omega\); in the paper under review the result is proved when \(G_\omega\) lies in a maximal subgroup from one of the Aschbacher families \(\mathcal C_2\) or \(\mathcal C_3\) [see M. Aschbacher, Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)].
Reviewer: Michael Bate (Oxford)
MSC:
| 20B15 | Primitive groups |
| 20G40 | Linear algebraic groups over finite fields |
| 20G05 | Representation theory for linear algebraic groups |
| 14H55 | Riemann surfaces; Weierstrass points; gap sequences |
| 14L30 | Group actions on varieties or schemes (quotients) |
Keywords:
finite classical groups; fixed point ratios; primitive permutation groups; monodromy groups; permutation representations; finite almost simple groups; maximal subgroups of classical groupsSoftware:
GAPReferences:
| [1] | Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math., 76, 469-514 (1984) · Zbl 0537.20023 |
| [2] | Aschbacher, M.; Seitz, G. M., Involutions in Chevalley groups over fields of even order, Nagoya Math. J., 63, 1-91 (1976) · Zbl 0359.20014 |
| [3] | Burness, T. C., Fixed point ratios in actions of finite classical groups, I, J. Algebra, 309, 1, 69-79 (2007) · Zbl 1128.20003 |
| [4] | Burness, T. C., Fixed point ratios in actions of finite classical groups, II, J. Algebra, 309, 1, 80-138 (2007) · Zbl 1128.20004 |
| [5] | Burness, T. C., Fixed point ratios in actions of finite classical groups, IV, J. Algebra, 314, 749-788 (2007) · Zbl 1133.20004 |
| [6] | Burness, T. C., Fixed point spaces in actions of classical algebraic groups, J. Group Theory, 7, 311-346 (2004) · Zbl 1071.20040 |
| [7] | The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4, 2004; The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4, 2004 |
| [8] | Gorenstein, D.; Lyons, R.; Solomon, R., The Classification of the Finite Simple Groups, Math. Surveys Monogr., vol. 40 (3) (1998), Amer. Math. Soc. · Zbl 0890.20012 |
| [9] | Kleidman, P. B.; Liebeck, M. W., The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser., vol. 129 (1990), Cambridge Univ. Press · Zbl 0697.20004 |
| [10] | Liebeck, M. W.; Shalev, A., Simple groups, permutation groups, and probability, J. Amer. Math. Soc., 12, 497-520 (1999) · Zbl 0916.20003 |
| [11] | F. Lübeck, private communication, 2005; F. Lübeck, private communication, 2005 |
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