# Fixed point ratios for finite primitive groups and applications

@article{Burness2021FixedPR,
title={Fixed point ratios for finite primitive groups and applications},
author={Timothy C. Burness and Robert M. Guralnick},
year={2021}
}
• Published 7 December 2021
• Mathematics

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. Let G be a ﬁnite group, let p be a prime and let Pr p ( G ) be the probability that two random p -elements of G commute. In this paper we prove that Pr p ( G ) > ( p 2 + p − 1) /p 3 if and only if
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A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is
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Preface Notational conventions 1. Introduction 2. Finite classical groups 3. Conjugacy classes 4. Subspace actions 5. Non-subspace actions 6. Low-dimensional classical groups Appendix A.
One of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group G of degree n, where G X An. We prove I G I < exp (4V'/ n log2 n) for the
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Abstract We improve a result of Liebeck and Saxl concerning the minimal degree of a primitive permutation group and use it to strengthen a result of Guralnick and Neubauer on generic covers of