Burness, Timothy C.
(2007)
Fixed point ratios in actions of finite classical groups, I.
*Journal of Algebra*, 309 (1), 69-79.
(doi:10.1016/j.jalgebra.2006.05.024).

## Abstract

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 ? 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,?) is one of a small number of known exceptions. Here fpr(x) denotes the proportion of points in ? 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.

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