The geometry of double loops
type: Item
instance of: scholarly article
title: The geometry of double loops
zbMATH Open document ID: 0664.51002
DOI: 10.1016/0001-8708(88)90018-7
author: A. Barlotti
author: Karl Strambach
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: By methods of classical geometry, the authors present an extensive study of not necessarily planar double loops \((Q,+,\cdot)\). A double loop is a set Q with two binary operations satisfying: \((Q,+)\) is a loop with neutral element 0, \((Q^*,\cdot)\) is a loop, \(Q^*:=Q\setminus \{0\}\), and \(0\cdot a=a\cdot 0=0\) for all \(a\in Q\). To every such double loop Q they assign a geometry \({\mathfrak N}\), called double net, which contains the 3-nets belonging to \((Q,+)\) and \((Q^*,\cdot)\) in a natural way [the authors, Adv. Math. 49, 1-105 (1983; Zbl 0518.20064)]. (In the classical case of a field Q, \({\mathfrak N}\) consists of the points in \(Q\times Q\), the horizontal and vertical lines (x,a) and (a,y), the transversal lines \(x+y=a,\) \(a\in Q\), and the hyperbolas \(xy=b\), \(b\in Q^*.)\) In general, \({\mathfrak N}\) is a rather weak geometry, but as in the paper cited above, the geometric approach turns out to be fruitful and leads to a wide variety of algebraic results on double loops. The main part of the paper is devoted to von Staudt's point of view, that is to characterizations of ``nice'' double loops Q via their group of projectivities \(\Theta\). Q is a commutative field if and only if \(\Theta\) is sharply 2-transitive on Q; a locally compact, not totally disconnected, topological double loop Q equals the field of real or complex numbers if and only if the pointwise stabilizer in \(\Theta\) of two different points is totally disconnected; a locally compact, 1- dimensional double loop Q is the field of the reals if and only if \(\Theta\) is locally compact; a finite double loop Q of order \(\neq 9,10,11,12,16\) is a field if and only if the pointwise stabilizer in \(\Theta\) of every five distinct points is trivial; in a finite double loop Q the loops \((Q,+)\) and \((Q^*,\cdot)\) are abelian groups if and only if \(\Theta\) has a regular abelian subgroup A, such that the stabilizer in \(\Theta\) of \(x\in Q\) has an abelian subgroup B regular on…
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 22A30
zbMATH DE Number: 4086411
zbMATH Keywords: 2-transitive permutation groups
zbMATH Keywords: topological double net
zbMATH Keywords: ordered double net
zbMATH Keywords: double loop
zbMATH Keywords: group of projectivities
zbMATH Keywords: topological double loop
zbMATH Keywords: free double loops
zbMATH Keywords: free double nets
zbMATH Keywords: ratios
zbMATH Keywords: conics
zbMATH Keywords: homomorphisms
reviewed by: Franz B. Kalhoff
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90018-7
OpenAlex ID: W2089143810
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