On the dimensions of automorphism groups of four-dimensional double loops
type: Item
instance of: scholarly article
title: On the dimensions of automorphism groups of four-dimensional double loops
zbMATH Open document ID: 0809.22005
DOI: 10.1007/BF02571700
author: Richard Bödi
published in: Mathematische Zeitschrift
publication date: 1995-03-29T00:00:00Z
full work available at URL: https://eudml.org/doc/174599
review text: In order to get started on Salzmann's classification program for compact projective planes, one needs stiffness results on the automorphism group, i.e., upper bounds on the dimensions of the stabilizers of suitable configurations. One hopes, for example, that the stabilizer of a quadrangle (which coincides with the automorphism group of a coordinatizing ternary) cannot be bigger than in the classical plane of the same dimension. This has been proved for ternaries of all possible finite dimensions except 4, where \(3 = \dim \text{Aut }\mathbb{H}\) is the desired bound, but only the bound 5 could be proved [\textit{H. Salzmann}, Geom. Dedicata 8, 139-161 (1979; Zbl 0465.51003)]. The lack of a sharp bound caused much additional work in the above-mentioned program. Now the author obtains the improved bound 4, and he does so under a weaker hypothesis; in fact, he does not use the ternary operation itself, but only addition and multiplication. Thus, he proves the bound for automorphism groups of 4-dimensional double loops. In addition to the methods used previously in this context, he employs Alexander duality in order to show that the group cannot be transitive on the complement of the fixed point set \(F\). Here, the possibility causing the trouble is that \(F\) might be 0-dimensional. Earlier, the author has shown that the complement of \(F\) is simply connected in this case [Arch. Math. 61, 291- 294 (1993; Zbl 0783.22001)]. This result is played off against Alexander duality. For 8-dimensional double loops, a similar result has been proved by the author [Monatsh. Math. 117, 1-16 (1994; Zbl 0796.22004)]. There, his bound is not quite as strong as the one known for ternaries.
reviewed by: Rainer Löwen
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 22D45
Mathematics Subject Classification ID: 51N25
Mathematics Subject Classification ID: 20N10
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 567419
zbMATH Keywords: dimensions of stabilizers
zbMATH Keywords: compact projective planes
zbMATH Keywords: coordinatizing ternary
zbMATH Keywords: automorphism groups
zbMATH Keywords: 4-dimensional double loops
zbMATH Keywords: Alexander duality
MaRDI profile type: Publication
cites work: Q5550843
cites work: On the embedding of zero-dimensional double loops in locally Euclidean double loops
cites work: Automorphism groups of locally compact connected double loops are locally compact
cites work: Q4081425
cites work: Q5541099
cites work: Q4039680
cites work: Q5643279
cites work: On Periodic Maps and the Euler Characteristics of Associated Spaces
cites work: Einige Sätze über topologische Gruppen
cites work: Q4156946
cites work: Q3972230
cites work: Dimension der Bahnen lokal kompakter Gruppen
cites work: Q3205756
cites work: On some types of topological groups
cites work: Topology and dimension of stable planes: On a conjecture of H. Freudenthal.
cites work: Q3884826
cites work: Toroid transformation groups on euclidean space
cites work: Q3229429
cites work: Q4069959
cites work: Topological planes
cites work: Homogene kompakte projektive Ebenen
cites work: Automorphismengruppen 8-dimensionaler Ternärkörper
cites work: Compact 8-dimensional projective planes with large collineation groups
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cites work: New results and old problems in finite transformation groups
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description: scientific article; zbMATH DE number 567419