Partial linear spaces with a rank 3 affine primitive group of automorphisms

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A partial linear space is a pair (Formula presented.) where (Formula presented.) is a non-empty set of points and (Formula presented.) is a collection of subsets of (Formula presented.) called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms (Formula presented.) of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when (Formula presented.) is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of (Formula presented.). Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.

Original languageEnglish
Pages (from-to)1011-1084
Number of pages74
JournalJournal of the London Mathematical Society
Issue number3
Early online date5 May 2021
Publication statusPublished - Oct 2021


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