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Partial linear spaces with a rank 3 affine primitive group of automorphisms
- Source :
- Journal of the London Mathematical Society
- Publication Year :
- 2021
- Publisher :
- Wiley, 2021.
-
Abstract
- A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ 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 $G$ 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 $G$ 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 $A\Gamma L_1(q)$. 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.<br />Comment: In this version, we have removed the assumption $V\leq H$ from 18.1 (old 13.2) and we have a new elementary proof of 10.10 (old 13.1). We have also reorganised some of the sections and made minor revisions throughout. 69 pages, 1 figure
- Subjects :
- Rank (linear algebra)
Group (mathematics)
General Mathematics
Linear space
010102 general mathematics
51E30, 05E18, 20B15, 05B25, 20B25
Group Theory (math.GR)
0102 computer and information sciences
Permutation group
Automorphism
01 natural sciences
Combinatorics
010201 computation theory & mathematics
Ordered pair
Line (geometry)
FOS: Mathematics
Mathematics - Combinatorics
Combinatorics (math.CO)
Affine transformation
0101 mathematics
Mathematics - Group Theory
Mathematics
Subjects
Details
- ISSN :
- 14697750 and 00246107
- Volume :
- 104
- Database :
- OpenAIRE
- Journal :
- Journal of the London Mathematical Society
- Accession number :
- edsair.doi.dedup.....233085073c6a00421136ef68fff6e94b