Timothy C. Burness

Fixed point ratios in actions of finite classical groups, II

This is the first in a series of four papers on fixed point ratios in actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr(x) |xG|−1/2 for all elements x ∈ G of prime order, or (G,Ω) is one of a small number of known exceptions. Here fpr(x) denotes the proportion of points in Ω which are fixed by x. In this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups. A further application concerning monodromy groups of covers of Riemann surfaces is also outlined.

On base sizes for algebraic groups

Let

Irreducible geometric subgroups of classical algebraic groups

G

Irreducible almost simple subgroups of classical algebraic groups

be a permutation group on a set

On Pyber's base size conjecture

\Omega

On the uniform spread of almost simple linear groups

. A subset of

Fixed point spaces in primitive actions of simple algebraic groups

\Omega

On the prime graph of simple groups

is a base for

On the number of prime order subgroups of finite groups

G

Base sizes for simple groups and a conjecture of Cameron

if its pointwise stabilizer is trivial; the base size of