Fixed point ratios in actions of finite exceptional groups of Lie type. (English) Zbl 1058.20001
Let \(G\) be a finite group acting transitively on a set \(\Omega\) and \(x\in\Omega\). The number
\[
\text{fpr}(x,\Omega)=\tfrac{\text{fix}_\Omega(x)}{|\Omega| },
\]
where \(\text{fix}_\Omega(x)\) is the number of fixed points of \(x\) on \(\Omega\), is called the fixed point ratio of \(x\). Fixed point ratios have been much studied in recent years, and applied to a number of different problems, particularly in the case where \(G\) almost simple.
In papers of D. Gluck and K. Magaard [Proc. Lond. Math. Soc., III. Ser. 71, No. 3, 547-584 (1995; Zbl 0865.20012)], M. W. Liebeck and A. Shalev [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)], D. Frohardt and K. Magaard [Geom. Dedicata 82, No. 1-3, 21-104 (2000; Zbl 0970.20003)] and R. M. Guralnick and W. M. Kantor [J. Algebra 234, No. 2, 743-792 (2000; Zbl 0973.20012)], upper bounds of fixed point ratios in classical groups \(G\) were obtained and applied to various problems.
M. W. Liebeck and J. Saxl [Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 266-314 (1991; Zbl 0696.20004)] obtained such a general upper bound of \(4/3q\) for groups of Lie type over \(F_q\) (with a few exceptions). These bounds were used by M. W. Liebeck and A. Shalev [loc. cit.] and by D. Frohardt and K. Magaard [Ann. Math. (2) 154, No. 2, 327-345 (2001; Zbl 1004.20001)] to prove the Guralnick-Thompson monodromy group conjecture. D. Frohardt and K. Magaard [Commun. Algebra 30, No. 2, 571-602 (2002; Zbl 1019.20011)] obtained better than \(4/3q\) bounds for exceptional groups of Lie type of rank at most 2.
In the present paper, the authors obtain stronger upper bounds for fixed point ratios of every group \(G\) with the socle \(L\) isomorphic to a finite simple exceptional group of Lie type. The proof of the main result is divided into several cases, giving upper bounds for \(\text{fpr}(x,\Omega)\) according to whether \(x\) is a semisimple or unipotent element, or \(x\) induces a non-diagonal outer automorphism of \(L\) of prime order, and also according to whether a point stabilizer is a parabolic or reductive subgroup. In many cases, the bounds given are close to best possible; in particular, this is the case for maximal parabolics. When a point stabilizer is a parabolic, character theory is used.
In other cases, the authors use their results [from Pac. J. Math. 205, No. 2, 339-391 (2002; see Zbl 1058.20039 below); correction ibid. 207, No. 2, 507 (2002)] on the analogous problem for algebraic groups and the results of M. W. Liebeck, J. Saxl and G. M. Seitz [Proc. Lond. Math. Soc., III. Ser. 65, No. 2, 297-325 (1992; Zbl 0776.20012)] and M. W. Liebeck and G. M. Seitz [Geom. Dedicata 35, No. 1-3, 353-387 (1990; Zbl 0721.20030), Trans. Am. Math. Soc. 350, No. 9, 3409-3482 (1998; Zbl 0905.20031)].
In papers of D. Gluck and K. Magaard [Proc. Lond. Math. Soc., III. Ser. 71, No. 3, 547-584 (1995; Zbl 0865.20012)], M. W. Liebeck and A. Shalev [J. Am. Math. Soc. 12, No. 2, 497-520 (1999; Zbl 0916.20003)], D. Frohardt and K. Magaard [Geom. Dedicata 82, No. 1-3, 21-104 (2000; Zbl 0970.20003)] and R. M. Guralnick and W. M. Kantor [J. Algebra 234, No. 2, 743-792 (2000; Zbl 0973.20012)], upper bounds of fixed point ratios in classical groups \(G\) were obtained and applied to various problems.
M. W. Liebeck and J. Saxl [Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 266-314 (1991; Zbl 0696.20004)] obtained such a general upper bound of \(4/3q\) for groups of Lie type over \(F_q\) (with a few exceptions). These bounds were used by M. W. Liebeck and A. Shalev [loc. cit.] and by D. Frohardt and K. Magaard [Ann. Math. (2) 154, No. 2, 327-345 (2001; Zbl 1004.20001)] to prove the Guralnick-Thompson monodromy group conjecture. D. Frohardt and K. Magaard [Commun. Algebra 30, No. 2, 571-602 (2002; Zbl 1019.20011)] obtained better than \(4/3q\) bounds for exceptional groups of Lie type of rank at most 2.
In the present paper, the authors obtain stronger upper bounds for fixed point ratios of every group \(G\) with the socle \(L\) isomorphic to a finite simple exceptional group of Lie type. The proof of the main result is divided into several cases, giving upper bounds for \(\text{fpr}(x,\Omega)\) according to whether \(x\) is a semisimple or unipotent element, or \(x\) induces a non-diagonal outer automorphism of \(L\) of prime order, and also according to whether a point stabilizer is a parabolic or reductive subgroup. In many cases, the bounds given are close to best possible; in particular, this is the case for maximal parabolics. When a point stabilizer is a parabolic, character theory is used.
In other cases, the authors use their results [from Pac. J. Math. 205, No. 2, 339-391 (2002; see Zbl 1058.20039 below); correction ibid. 207, No. 2, 507 (2002)] on the analogous problem for algebraic groups and the results of M. W. Liebeck, J. Saxl and G. M. Seitz [Proc. Lond. Math. Soc., III. Ser. 65, No. 2, 297-325 (1992; Zbl 0776.20012)] and M. W. Liebeck and G. M. Seitz [Geom. Dedicata 35, No. 1-3, 353-387 (1990; Zbl 0721.20030), Trans. Am. Math. Soc. 350, No. 9, 3409-3482 (1998; Zbl 0905.20031)].
Reviewer: Anatoli Kondrat’ev (Ekaterinburg)
MSC:
| 20B15 | Primitive groups |
| 20G40 | Linear algebraic groups over finite fields |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20C33 | Representations of finite groups of Lie type |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
