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Theorie der verallgemeinerten kinematischen Räume. (Theory of generalized kinematic spaces). (German) Zbl 0636.51012
Incidence spaces (G,\({\mathfrak G})\) are considered with regular sets of collineations, the so-called incidence loops. To any metric plane there belongs such a structure namely a kinematic space which is embedded into a projective space P such that the complement Q is a 2-set: \(Q\subset P\) is called a 2-set if for any line \(X\subset P\) one has \(| X\cap Q| \leq 2\) or \(X\subset Q\). In the first part of the paper a survey is given on those 2-sets, especially on s-homogeneous ones. Q is called regular, if \(P\setminus Q\) is an incidence loop. It is shown that knot ovals Q with \(| Q| \neq 4\) are not regular and that an ovoid in a 3-dimensional projective space can not be a 2-set of an incidence group. So, there is the existence problem: Does any regular Q contain at least one line?
In the second part properties of incidence groups are generalized to incidence loops. Of special interest are the Moufang incidence loops with parallelism. Furthermore it is shown that any linear fibered Moufang incidence loop with a 2-set as a complement in a projective space is two- sided, and a classification of all those incidence loops is given.
Reviewer: H.Hotje

MSC:
51J05 General theory of incidence groups
20N05 Loops, quasigroups
51A15 Linear incidence geometric structures with parallelism
51A45 Incidence structures embeddable into projective geometries
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[1] Andre, J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60 (1954), 156–186 · Zbl 0056.38503 · doi:10.1007/BF01187370
[2] Andre, J. Über Parallelstrukturen. I., II. Math. Z. 76 (1961), 85–102 und 155-163 · Zbl 0099.15401 · doi:10.1007/BF01210963
[3] Baer, R.: Partitionen endlicher Gruppen. Math. Z. 75 (1961), 333–372 · Zbl 0103.01404 · doi:10.1007/BF01211032
[4] BrÖcker, L.: Kinematische Räume. Geometriae Dedicata 1 (1973), 241–278 · Zbl 0249.50014 · doi:10.1007/BF00147523
[5] Bruck, R.H.: A survey of binary systems. Berlin-Göttingen- Heidelberg 1958 · Zbl 0081.01704
[6] Coxeter, H.S.M.: Twelve points in PG(5,3) with 95040 self-transformations. Proc. Royal Soc. (A) 247 (1958), 279–293 · Zbl 0082.36207 · doi:10.1098/rspa.1958.0184
[7] Glauberman, G.: On Loops of Odd Order. J. Algebra 1 (1966), 374–396 · Zbl 0123.01502 · doi:10.1016/0021-8693(64)90017-1
[8] Heise, W.: Bericht über \(\kappa\)-affine Geometrien. J. of Geometry 1 (1971), 197–224 · Zbl 0228.50033 · doi:10.1007/BF02150272
[9] Karzel, H.: Gruppentheoretische Begründung metrischer Geometrien. Vorlesungsausarbeitung von G. Graumann, Hamburg 1963
[10] Karzel, H. Bericht über projektive Inzidenzgruppen. Jber. Deutsch. Math.-Verein. 67 (1964), 58–92 · Zbl 0131.19101
[11] Karzel, H. Spiegelungsgruppen und absolute Gruppenräume. Abh. Math. Sem. Univ. Hamburg 35 (1971), 141–163 · Zbl 0212.52202 · doi:10.1007/BF02993621
[12] Karzel, H. Kinematic spaces. Symposia Mathematica. Ist. Naz. di Alta Matematica 11. (1973), 413–439
[13] Karzel, H. und Kist, G.: Kinematic algebras and their geometries. Proc. of the NATO Advanced Study Institute on Rings and Geometry. Hsg. von R.Kaya, P.Plaumann, K.Strambach. Dordrecht (1985), 437-509 · Zbl 0598.51012
[14] Karzel, H. Finite porous incidence groups and s-homogeneous 2-sets. Symposia Mathematica, 1st. Naz. di Alta Matematica 28 (1986), 89–111 · Zbl 0599.51020
[15] Karzel, H. und Marchi, M.: Planar fibered incidence groups. J. of Geometry 20 (1983), 192–201 · Zbl 0515.51014 · doi:10.1007/BF01918009
[16] Karzel, H. Und Maxson, C.J.: Kinematic spaces with dilatations. J. of Geometry 22 (1984), 196–201 · Zbl 0537.51022 · doi:10.1007/BF01222846
[17] Karzel, H. Fibered p-groups. Abh. Math. Sem. Univ. Hamburg 56 (1986), 1–9 · Zbl 0618.20016 · doi:10.1007/BF02941502
[18] Karzel, H. Und Pieper, I.: Bericht über geschlitzte Inzidenzgruppen. Jber. Deutsch. Math.-Verein. 72 (1970), 70–114 · Zbl 0202.51001
[19] Karzel, H. Kroll, H.-J. und SÖrensen, K.: Invariante Gruppenpartitionen und Doppelräume. J. Reine Angew. Math. 262/263 (1973), 153–157 · Zbl 0265.50003
[20] Karzel, H. SÖrensen, K. und Windelberg, D.: Einführung in die Geometrie. Göttingen 1973 · Zbl 0248.50001
[21] Kegel, O.H.: Lokal endliche Gruppen mit nicht-trivialer Partition. Arch. Math. 13 (1962), 10–28 · Zbl 0106.24701 · doi:10.1007/BF01650044
[22] Kist, G.: Kinematische punktiert-affine Inzidenzgruppen. Separatum aus: Beiträge zur Geometrischen Algebra. Hsg. von H.J. Arnold, W.Benz, H.Wefelscheid, Basel (1977), 179-183
[23] Kist, G. Projektiver Abschlu\(\beta\) 2-gelochter Räume. Res. d. Math. 3. (1980), 192–211 · Zbl 0477.51004 · doi:10.1007/BF03323356
[24] Kist, G. Theorie der verallgemeinerten kinematischen Räume. Habilitationsschrift TU München (1980) · Zbl 0477.51004
[25] Kroll, H.-J.: Bestimmung aller projektiven Doppelräume. Abh.Math. Sem. Univ. Hamburg 44 (1975), 139–142 · Zbl 0317.50005 · doi:10.1007/BF02992953
[26] Robinson, D.A.: Bol Loops. Trans. Amer. Math. Soc. 123 (1966), 341–354 · Zbl 0163.02001 · doi:10.1090/S0002-9947-1966-0194545-4
[27] SchrÖder, E.M.: Kennzeichnung und Darstellung kinematischer Räume metrischer Ebenen. Abh. Math. Sem. Univ. Hamburg 39. (1973), 184–230 · Zbl 0264.50003 · doi:10.1007/BF02992831
[28] SchrÖder, E.M. Zur Theorie subaffiner Inzidenzgruppen. J. of Geometry 3. (1973), 31–69 · Zbl 0245.50006 · doi:10.1007/BF01949704
[29] WÄhling, H.: Projektive Inzidenzgruppoide und Fastalgebren. J. of Geometry 9. (1977), 109–126 · Zbl 0351.50009 · doi:10.1007/BF01918063
[30] Wielandt,H.: Finite Permutation Groups. New York-London 1964 · Zbl 0138.02501
[31] Zick,W.: Geschlitzte Inzidenzgruppoide. Dissertation, Hannover 1979
[32] Zizioli,E.: A Class of Fibered Loops. In diesem Proceedingsband. · Zbl 0643.20048
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