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SEMINAR: Groups and Combinatorics Seminar: Pairwise transitive 1-designs

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Today's date is Friday, March 29, 2024
Groups and Combinatorics Seminar: Pairwise transitive 1-designs Other events...
Time and place: 15:00 Friday 23 October in Weatherburn LT.

Speaker: Adrian Petersen (University of Western Australia)

Title: Pairwise transitive 1-designs

Abstract: Design theory has extensive interactions with other areas of mathematics, especially group theory and graph theory. Indeed, the problem investigated here arises from the study of a particular family of graphs, called locally s-distance transitive graphs. In this dissertation we investigate the relationship between a 1-design and its automorphism group, specifically we aim to classify pairwise transitive 1-designs. We say that a design is G-pairwise transitive if a group G of automorphisms of the design is transitive on the following ordered sets: collinear point pairs, non-collinear point pairs, intersecting block pairs, non-intersecting block pairs, flags, and anti-flags. Some of the transitivity conditions have been studied previously for 1-designs, but never have they been imposed all at once. In a way, a stronger symmetry property could not be imposed on a 1-design, as we require transitivity on every possible ordered pair of a design.

We draw inspiration from work by Devillers and Praeger, who classified the less general class of pairwise transitive 2-designs. This classification relied on the 2-transitive permutation groups, which are all known. In our case an original classification scheme is developed, which says that rank 3 permutation groups are needed to construct pairwise transitive 1-designs. Rank 3 permutation groups can be either primitive or imprimitive. The former case is very well understood; all primitive rank 3 actions are classified, and we take advantage of this classification. The latter case is not well understood. The primitive rank 3 groups can be further subdivided into the almost simple, affine, and grid type groups. This dissertation presents a partial classification of G-pairwise transitive strict 1-designs for primitive almost simple G.
Contact Gabriel Verret <[email protected]>
Start Fri, 23 Oct 2015 15:00
End Fri, 23 Oct 2015 16:00
Submitted by Gabriel Verret <[email protected]>
Last Updated Wed, 21 Oct 2015 10:47
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