The generalized André systems \(A(F,\Gamma,(g_ i),(f_ i),\in)\)
type: Item
instance of: scholarly article
title: The generalized André systems \(A(F,\Gamma,(g_ i),(f_ i),\in)\)
zbMATH Open document ID: 0695.51001
DOI: 10.1007/BF02941379
author: A. Caggegi
author: Armin Herzer
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1988-01-01T00:00:00Z
review text: Given a field \(Q=(Q,+,\cdot)\), a map \(\phi\) : \(Q^*\to Aut Q\); \(x\to \phi_ x\) \((Q^*:=Q\setminus \{0\}\), Aut Q: group of automorphisms of Q) is called an automorphic derivation map of Q, if \(\phi_ 1=id\) and for every pair (a,b) of elements of \(Q^*\) there exists one and only one \(x\in Q^*\) with \(x\phi_ x(a)=b\). Defining \(a\circ b:=a\phi_ a(b)\) if \(a\neq 0\) and \(0\circ b=0\), \(Q^{\phi}:=(Q,+,\circ)\) becomes a quasifield, often called a generalized André-system, i.e. \((Q,+)\) is an abelian group, \((Q^*,\circ)\) is a loop with identity 1, \(0\circ a=0\) and \(a\circ (b+c)=a\circ b+a\circ c\) for all a,b,c\(\in Q\). A quasifield is said to be planar, if the equation \(a\circ x=b\circ x+c\) has a solution whenever \(a\neq b\). Planarity is necessary and sufficient for quasifields to coordinatize translation planes. The authors call planar quasifields simply quasifields, while not necessarily planar quasifield are attributed weak. They give a slightly circumstantial proof of the fact that \(Q^{\phi}\) is planar and that the fixed field K of \(\phi (Q^*)\) in Q equals the kernel of Q if Q is algebraic over K. An extensive class of automorphic derivations is defined. After some remarks how to construct explicitly derivations of this kind, the authors proceed in showing that the following known classes of quasifields are contained in their class: Lüneburg's generalization of André- quasifields, \(Q^{\phi}\) whenever Q/K is a cyclic extension of prime power degree, the finite Dickson-nearfields and the class of finite quasifields \(Q_ g\) constructed by Foulser. The last section deals with the collineation groups of the corresponding translation planes in certain special cases.
reviewed by: Hubert Kiechle
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 12K99
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 4138654
zbMATH Keywords: quasifield
zbMATH Keywords: generalized André-system
MaRDI profile type: Publication
cites work: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe
cites work: Piani di Andre generalizzati
cites work: On generalized André planes of dimension \(2(2n+1)\)
cites work: Some new generalized André systems
cites work: Q5545191
cites work: A generalization of André's systems
cites work: Q3943560
cites work: Q3891315
cites work: A Question of Foulser on λ-Systems of Characteristic Two
cites work: Eine Klasse unendlicher verallgemeinerter Andre-Ebenen
description: scientific article; zbMATH DE number 4138654