Abstract
The involution fixity ifix(G) of a permutation group G of degree n is the
maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if T is the socle of such a group, then either ifix(T) > n1/3, or ifix(T) = 1 and T = 2B2(q) is a Suzuki group in its natural 2-transitive action of degree n = q 2 + 1. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with ifix(T) 6 n 4/9. This extends recent work of Liebeck and Shalev, who established the bound ifix(T) > n1/6 for every almost simple primitive group
of degree n with socle T (with a prescribed list of exceptions). Finally, by combining our results with the Lang-Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.
Original language | English |
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Pages (from-to) | 411-466 |
Number of pages | 56 |
Journal | International Journal of Algebra and Computation |
Volume | 28 |
Issue number | 3 |
Early online date | 10 Apr 2018 |
DOIs | |
Publication status | Published - May 2018 |
Keywords
- Finite exceptional groups
- involution fixity
- primitive actions