The isomorphism problem for abelian projective planes
type: Item
instance of: scholarly article
title: The isomorphism problem for abelian projective planes
zbMATH Open document ID: 1154.51001
author: Dieter Jungnickel
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 2008-08-20T00:00:00Z
review text: Let \(G\) be a cyclic group of order \(v\), and let \(\Pi\) and \(\Pi'\) be two cyclic incidence structures with point set \(V=G\), i.e., \(G\) acts by right translation as an automorphism group of both \(\Pi\) and \(\Pi'\). Then \(\Pi\) and \(\Pi'\) are called \textit{multiplier equivalent} if some automorphism \(\alpha\) of \(G\) induces an isomorphism between \(\Pi\) and \(\Pi'\). Then \(\alpha\) has the form \(\alpha:x\mapsto tx\), where \(t\) is a positive integer with \(\gcd(v,t)=1\). Moreover, if \(G\) is an arbitrary finite group, then \(\Pi\) and \(\Pi'\) are called \textit{equivalent with respect to \(G\)} if some group isomorphism \(\alpha\) of \(G\) induces an isomorphism between \(\Pi\) and \(\Pi'\). This article proves that if \(G\) is an abelian group of order \(n^2+n+1\), then two \(G\)-invariant projective planes of order \(n\) are isomorphic if and only if they are equivalent with respect to \(G\). As a corollary, this result shows that two cyclic projective planes of order \(n\) are isomorphic if and only if they are multiplier equivalent, that is, if and only if the associated difference sets in \(Z_{n^2+n+1}\) are equivalent. The motivation of the author is the Bays-Lambossy theorem. The Bays-Lambossy theorem states that if \(p\) is a prime, and \(\Pi\) and \(\Pi'\) are two cyclic incidence structures on \(p\) points, where the point sets of both structures are the cyclic group \(Z_p\), then if \(\Pi\) and \(\Pi'\) are isomorphic, then they are multiplier equivalent. In this way, the author obtains an equivalent to the Bays-Lambossy theorem for finite projective planes.
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51E10
zbMATH DE Number: 5314403
zbMATH Keywords: Projective planes
zbMATH Keywords: Bays-Lambossy theorem
zbMATH Keywords: isomorphism problem
reviewed by: Leo Storme
MaRDI profile type: Publication
full work available at URL: s00200-008-0070-4
OpenAlex ID: W1989787556
cites work: Q4718791
cites work: Q5545191
cites work: Q4370894
cites work: Finite projective planes with abelian transitive collineation groups
cites work: Uniqueness of some cyclic projective planes
cites work: Multipliers and generalized multipliers of cyclic objects and cyclic codes
cites work: Classification of 2-transitive symmetric designs
cites work: Sur une maniere de différencier les fonctions cycliques d'une forme donnée
cites work: On the isomorphism problem for cyclic combinatorial objects
cites work: A solution of the isomorphism problem for circulant graphs
cites work: Endliche zyklische Ebenen
cites work: Isomorphism problem for relational structures with a cyclic automorphism
cites work: A Theorem in Finite Projective Geometry and Some Applications to Number Theory
DOI: 10.1007/S00200-008-0070-4
description: scientific article; zbMATH DE number 5314403