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Rank three innately transitive permutation groups and related $2$-transitive groups

Anton A. Baykalov, Alice Devillers and Cheryl E. Praeger

Vol. 20 (2023), No. 2-3, 135–175
Abstract

The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most 3 to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple 2-transitive groups called special pairs.

Dedicated to the memory of Jacques Tits

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Keywords
rank 3, permutation group, partial linear space, innately transitive, 2-transitive
Mathematical Subject Classification
Primary: 20B05, 20B10, 20B20
Milestones
Received: 22 August 2022
Revised: 5 November 2022
Accepted: 4 December 2022
Published: 13 September 2023
Authors
Anton A. Baykalov
Centre for the Mathematics of Symmetry and Computation
Department of Mathematics and Statistics
The University of Western Australia
Perth
Australia
Alice Devillers
Centre for the Mathematics of Symmetry and Computation
Department of Mathematics and Statistics
The University of Western Australia
Perth
Australia
Cheryl E. Praeger
Centre for the Mathematics of Symmetry and Computation
Department of Mathematics and Statistics
The University of Western Australia
Perth
Australia