On abelian projective planes
type: Item
instance of: scholarly article
title: On abelian projective planes
zbMATH Open document ID: 0571.51005
DOI: 10.1007/BF01237865
author: Dina Ghinelli-Smit
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: In this paper the author uses character theory and the notion of discriminant to prove the following nonexistence condition for projective planes of non square order n, with an abelian Singer group G: ''If h is the order of a subgroup of G, and p is a prime divisor of n, then \(h(n^ 2+n+1)(-1)^{[(n^ 2+n+1)-h]/2h}\) must be a square in the ring of p- adic numbers''. This condition is stronger than all known results in this direction. As an application the author considers the special case \(n\equiv 1,2(mod 4).\) For \(h=1\), the above condition yields that if n is not a square, then -1 must be a square module each odd prime divisor of n, and not only for odd prime divisors of the square free part of n (as one would get from the Bruck-Ryser-Chowla theorem). In general, the author deduces even more, namely: ''If \(n\equiv 1,2(mod 4),\) then the order h of each subgroup of an abelian Singer group must be a square modulo every odd prime divisor of n, or n must be a square''.
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 3911166
zbMATH Keywords: projective planes
zbMATH Keywords: abelian Singer group
MaRDI profile type: Publication
cites work: Automorphisms and Generalized Incidence Matrices of Point-Divisible Designs
cites work: Cyclic Incidence Matrices
cites work: On the geometry of planar difference sets
cites work: Collineations and Generalized Incidence Matrices
cites work: Q3856855
description: scientific article; zbMATH DE number 3911166