On flag-transitive 2-(v,k,2) designs

2020/01/14 by Alice Devillers, Hongxue Liang, Devillers, Alice +5 · 3 citations
Computer Science · Engineering · Mathematics · #05B05 #05B25 #20B25 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2001.04728

openalex publication_date 2020/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n ≥ 3. Alongside this analysis, we give a construction for a flag-transitive 2-(v,k-1,k-2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line. Taking the design of points and lines of the projective space PG(n-1,3) as input to this construction yields a G-flag-transitive 2-(v,3,2) design where G has socle PSL(n,3) and v=(3ⁿ-1)/2. Apart from these designs, our PSL-classification yields exactly one other example, namely the complement of the Fano plane.

Citations

• Linear spaces with flag-transitive automophism groups

Cited by

• Large maximal subgroups of almost simple classical groups
• Flag-transitive point-primitive quasi-symmetric 2-designs and exceptional groups of Lie type
• On flag-transitive automorphism groups of 2-designs with λ prime

Related

• Flag-transitive 2-(v,k,λ) designs with λ≥ (r,λ)²
• A Classification of the flag-transitive 2-(v,k,2) designs
• Analysing 2-(v,k,2) designs admitting a flag-transitive almost simple automorphism group with socle PSL(2,q) by means of conics and hyperovals of PG(2,q)
• 2-(v,k,3) designs admitting an almost simple, flag-transitive automorphism group with socle PSL(2,q)
• Classification of flag-transitive 2-(v, k, λ) designs with alternating group A_n(n ≤ 10) as socle