2011/08/01 by Massimo Giulietti, Gabor Korchmaros, Giulietti, Massimo +6
Computer Science · Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Coding theory and cryptography #Combinatorics #Combinatorics (math.CO) #Complement (music) #Conjugacy class #Cyclic group #FOS: Mathematics #Finite Group Theory Research #Finite group #Geometry #Group (periodic table) #Invariant (physics) #Mathematical physics #Mathematics #Order (exchange) #Physics #Projective plane #math.CO
paper · pdf · doi:10.48550/arxiv.1108.0358
published in arXiv (Cornell University) (Cornell University) · 10 pages
openalex publication_date 2011/08/01 · arxiv created 2011/08/31 · arxiv updated 2011/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
For q=pr with a prime p≥ 7 such that q ≡ 1 or 19\pmod 30, the desarguesian projective plane PG(2,q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A6 of degree 6. For a projectivity group Γ≅ A6 of PG(2,q), we investigate the geometric properties of the (unique) Γ-orbit O of size 90 such that the 1-point stabilizer of Γ in \mathcal O is a cyclic group of order 4. Here \mathcal O lies either in PG(2,q) or in PG(2,q2) according as 3 is a square or a non-square element in GF(q). We show that if q≥ 349 and q≠ 421, then \mathcal O is a 90-arc, which turns out to be complete for q=349, 409, 529, 601,661. Interestingly, \mathcal O is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.