Symmetries of biplanes

Seyed Hassan Alavi, Ashraf Daneshkhah, Cheryl E. Praeger

Research output: Contribution to journalArticle

Abstract

In this paper, we first study biplanes D with parameters (v, k, 2), where the block size k∈ { 13 , 16 }. These are the smallest parameter values for which a classification is not available. We show that if k= 13 , then either D is the Aschbacher biplane or its dual, or Aut(D) is a subgroup of the cyclic group of order 3. In the case where k= 16 , we prove that | Aut(D) | divides 2 7· 3 2· 5 · 7 · 11 · 13. We also provide an example of a biplane with parameters (16, 6, 2) with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.

Original languageEnglish
JournalDesigns, Codes, and Cryptography
DOIs
Publication statusE-pub ahead of print - 7 Aug 2020

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