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- Title
On the involution fixity of exceptional groups of Lie type.
- Authors
Burness, Timothy C.; Thomas, Adam R.
- 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=q2+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)≤n4/9. 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.
- Subjects
GROUP theory; PARAMETER estimation; NUMBER theory; APPROXIMATION theory; RATIONAL choice theory
- Publication
International Journal of Algebra & Computation, 2018, Vol 28, Issue 3, p411
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196718500200