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Gruppenuniversalität und Homogenisierbarkeit. (German) Zbl 0588.03021

The authors use model theory to prove that, for very general classes \({\mathcal C}\) of structures: (1) every group G is the full automorphism group of some \(C\in {\mathcal C}\), (2) every \(C\in {\mathcal C}\) may be embedded in a homogeneous \(H\in {\mathcal C}\). The results unify numerous results from the literature. The authors discuss the applicability of the results to groups, projective planes, affine planes, (m,n)-planes, (k,m,n)- geometries, generalized n-corners (or vertices), k-nets, (k,n)-Steiner systems, LP-spaces, covering geometries, ”magmas” of type \(\tau\), \(\theta\)-algebras of type \(\tau\), commutative loops and quasigroups, commutative algebras of type \(\tau\) over a commutative field, and commutative division algebras over a commutative field.
Reviewer: J.Cannon

MSC:

03C99 Model theory
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
51Exx Finite geometry and special incidence structures
20A15 Applications of logic to group theory
20B27 Infinite automorphism groups
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[1] André, J., Über geometrische Strukturen, die zu Permutationsgruppen gehören, Abh. Math. Sem. Univ. Hamburg, 44, 203-221 (1976) · Zbl 0318.20002
[2] André, J., Nichtkommutative Geometrie und verallgemeinerte Hughes-Ebenen, Math. Zeitschr., 177, 449-462 (1981) · Zbl 0444.51003
[3] Babai, L., Infinite digraphs with given regular automorphism groups, J. Comb. Th., B 25, 26-46 (1978) · Zbl 0392.05032
[4] Babai, L., Vector representable matroids of given rank with given automorphism group, Discrete Math., 24, 119-125 (1978) · Zbl 0395.05023
[5] L.Babai,On the abstract group of automorphisms, in: «Combinatorics» (herausg. von H. N. V.Temperley), London Math. Soc. Lect. Note Series,52 (1981), pp. 1-40. · Zbl 0467.05031
[6] Babai, L.; Duffus, D., Dimension and automorphism groups of lattices, Alg. Universalis, 12, 279-289 (1981) · Zbl 0495.06002
[7] Babai, L.; Frankl, P., Infinite quasigroups with given regular automorphism groups, Alg. Universalis, 8, 310-319 (1978) · Zbl 0394.20057
[8] Babai, L.; Imrich, W., Tournaments with given regular group, Aequationes Math., 19, 232-244 (1979) · Zbl 0422.05034
[9] Bachmann, O., Embeddings and collineation groups of projective planes, Archiv der Math., 29, 129-135 (1977) · Zbl 0359.50002
[10] Barlotti, A., Sulle m-strutture di Möbius, Rend. Ist. di Mat. Univ. Trieste, 1, 35-46 (1969) · Zbl 0183.49703
[11] Barlotti, A.; Strambach, K., The geometry of binary systems, Advances in Math., 49, 1-105 (1983) · Zbl 0518.20064
[12] Barwise, J., Handbook of Mathematical Logic (1977), Amsterdam-New York-Oxford: North Holland, Amsterdam-New York-Oxford
[13] Benz, W., Vorlesungen über Geometrie der Algebren (1973), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0258.50024
[14] Birkhoff, G., On the structure of abstract algebras, Proc. Cambridge Phil. Soc., 31, 433-454 (1935) · Zbl 0013.00105
[15] Birkhoff, G., Sobre los grupos de automorfismos, Revista Unión Mat. Argentina, 11, 155-157 (1945)
[16] P. J.Cameron,Orbits, enumeration and colouring, in: «Combinatorial Mathematics IX», Proc. Brisbane, Australia, 1981; Lecture Notes in Math., Berlin-Heidelberg-New York, Springer,952 (1982), pp. 34-66.
[17] Cayley, A., On the theory of groups, Proc. London Math. Soc., 9, 126-133 (1878) · JFM 10.0104.01
[18] Chang, C. C.; Keisler, H. J., Model Theory (1973), Amsterdam-London: North Holland, Amsterdam-London · Zbl 0276.02032
[19] Chvátal, V.; Hell, P.; Kučera, L.; Nešetřil, J., Every finite graph is a full subgraph of a rigid graph, J. Comb. Th., 11, 284-286 (1971) · Zbl 0231.05107
[20] Civolani, N., Hyperfree extensions of partial Klingenberg planes, Geometriae Dedicata, 9, 467-475 (1980) · Zbl 0455.51004
[21] Clark, D. M.; Kraus, P. M.; Gandy, R.; Hyland, M., Relatively homogeneous structures, «Logic Colloquium », 1976, 255-282 (1977), Amsterdam-London: North Holland, Amsterdam-London
[22] Coxeter, H. S. M.; Moser, W. O. J., Generators and relations for discrete groups (1957), Berlin: Springer, Berlin · Zbl 0077.02801
[23] Dembowski, P., Finite Geometries (1968), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York
[24] Dénes, J.; Keedwell, A. D., Latin Squares and Their Applications (1974), Budapest: Akadémiai Kiado, Budapest · Zbl 0283.05014
[25] Evans, T., Embedding theorems for multiplicative systems and projective geometries, Proc. American Math. Soc., 3, 614-620 (1952) · Zbl 0047.02102
[26] Fried, M., A note on automorphism groups of algebraic number fields, Proc. American Math. Soc., 80, 386-388 (1980) · Zbl 0492.12007
[27] Fried, E.; Kollár, J., Automorphism groups of algebraic number fields, Math. Zeitschr., 163, 121-123 (1978) · Zbl 0391.12005
[28] E.Fried - J.Kollár,Automorphism groups of fields, in: «Universal Algebra» (herausg. von E. T.Schmidt u.a.), Coll. Math. soc. J. Bolyai,24 (1981). · Zbl 0501.20023
[29] Frucht, R., Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math., 6, 239-250 (1938) · Zbl 0020.07804
[30] Frucht, R., Graphs of degree 3 with a given abstract group of automorphisms, Canad. J. Math., 1, 365-378 (1949) · Zbl 0034.25802
[31] Frucht, R., Lattices with given abstract group of automorphisms, Canad. J. Math., 2, 417-419 (1950) · Zbl 0039.02403
[32] Fuhrken, G., On automorphisms of algebras with a single unary operation, Portugaliae Math., 32, 49-52 (1973) · Zbl 0254.08003
[33] Geyer, W. D.; Dold, A.; Eckmann, B., Invarianten binärer Formen, Classification of Algebraic Varieties and Compact Complex Manifolds, 36-69 (1974), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York
[34] Geyer, W. D., Jede endliche Gruppe ist Automorphismengruppe einer endlichen Erweiterung K¦ℚ, Archiv der Math., 41, 139-142 (1983) · Zbl 0515.12005
[35] Gould, M., A note on automorphisms of groupoids, Alg. Universalis, 2, 54-56 (1972) · Zbl 0248.20071
[36] Gould, M., Automorphism and subalgebra structure in algebras of finite type, Alg. Universalis, 2, 369-374 (1972) · Zbl 0273.08007
[37] Grätzer, G., Universal Algebra (1968), Princeton: Van Nostrand, Princeton
[38] Grätzer, G.; Sichler, J., On the endomorphism semigroup (and category) of bounded lattices, Pacific J. Math., 35, 639-647 (1970) · Zbl 0208.02602
[39] R. J.Greechie,Finite groups as automorphism groups of orthocomplemented projective planes, Preprint. · Zbl 0377.05010
[40] J.de Groot,Automorphism groups of rings, Internat. Congr. Math., Edinburgh, (1958), p. 18.
[41] De Groot, J., Groups represented by homeomorphism groups I, Math. Annalen, 138, 80-102 (1959) · Zbl 0087.37802
[42] Grünwald, J., Über duale Zahlen und ihre Anwendung in der Geometrie, Monatsh. Math. Phys., 17, 81-136 (1906) · JFM 37.0486.01
[43] Halin, R., Graphentheorie (1980), Darmstadt: Wiss. Buchgesellschaft, Darmstadt
[44] Hall, M., Protective planes, Trans. American Math. Soc., 54, 229-277 (1943) · Zbl 0060.32209
[45] Hartshorne, R., Algebraic Geometry (1977), New York-Heidelberg-Berlin: Springer, New York-Heidelberg-Berlin
[46] Z.Hedrlín,On endomorphisms of graphs and their homomorphic images, in: Proof Techniques in Graph Theory (herausg. von F.Harrary), Academic Press, 1969. · Zbl 0199.27503
[47] Hedrlín, Z.; Lambek, J., How comprehensive is the category of semigroups?, Journal of Algebra, 11, 195-212 (1969) · Zbl 0206.02505
[48] Hedrlín, Z.; Mendelsohn, E., The category of graphs with given subgraph, Canad. J. Math., 21, 1506-1517 (1969) · Zbl 0196.03702
[49] Hedrlín, Z.; Pultr, A., Relations (graphs) with given infinite semigroups, Monatsh. Math., 68, 421-425 (1964) · Zbl 0139.24802
[50] Hedrlín, Z.; Pultr, A., Symmetric relations (undirected graphs) with given semigroups, Monatsh. Math., 69, 318-322 (1965) · Zbl 0139.24803
[51] Heise, W., Bericht über ϰ-affine Geometrien, J. of Geometry, 1, 197-224 (1971) · Zbl 0228.50033
[52] Heise, W.; Karzel, H., Laguerre- und Minkowski-m-Strukturen, Rend. Ist. Mat. Univ. Trieste, 4, 139-147 (1972) · Zbl 0248.50028
[53] Heise, W.; Sörensen, K., Freie Minkowski-Ebenenerweiterungen, J. of Geometry, 3, 1-4 (1973) · Zbl 0252.50031
[54] Higman, G.; Neumann, B. H.; Neumann, H., Embedding theorems for groups, J. London Math. Soc., 24, 247-254 (1949) · Zbl 0034.30101
[55] Izbicki, H., Reguläre Graphen 3., 4.und 5.Grades mit vorgegebenen abstrakten Automorphismengruppen, Farbenzahlen und Zusammenhängen, Monatsh. Math., 61, 42-50 (1957) · Zbl 0077.17101
[56] Jónsson, B., Homogeneous universal relational systems, Math. Scand., 8, 137-142 (1960) · Zbl 0173.00505
[57] Jónsson, B., Algebraic extensions of relational systems, Math. Scand., 11, 179-205 (1962) · Zbl 0201.34403
[58] Jónsson, B., Algebraic structures with prescribed automorphism groups, Coll. Math., 19, 1-4 (1968) · Zbl 0153.33603
[59] Karzel, H., Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom, Abh. Math. Sem. Univ. Hamburg, 32, 191-206 (1968) · Zbl 0162.24101
[60] Karzel, H.; Kroll, H.-J.; Plaumann, P.; Strambach, K., Perspectivities in circle geometries, Geometry- von Staudt’s Point of View, 51-99 (1981), Dortrecht-Boston-London: Reidel, Dortrecht-Boston-London
[61] Kegel, O. H.; Schleiermacher, A., Amalgams and embeddings of projective planes, Geometriae Dedicata, 2, 379-395 (1973) · Zbl 0271.50017
[62] Keisler, H. J., Model Theory for Infinitary Logic (1971), Amsterdam-London: North Holland, Amsterdam-London
[63] Kerby, W.; Wefelscheid, H., Über eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur, Abh. Math. Sem. Univ. Hamburg, 37, 225-235 (1972) · Zbl 0258.17010
[64] Klingenberg, W., Projektive Geometrien mit Homomorphismus, Math. Ann., 132, 180-200 (1956) · Zbl 0073.36404
[65] König, D., Theorie der endlichen und unendlichen Graphen (1936), Leipzig: Akademischer Verlag, Leipzig · JFM 62.0654.05
[66] Lange, H., Über die Modulschemata der Kurven vom Geschlecht 2mit 1, 2oder 3Weierstraβpunkten, J. reine und angew. Math., 277, 27-36 (1975) · Zbl 0317.14011
[67] Marley, M.; Vaught, R., Homogeneous universal models, Math. Scand., 11, 37-67 (1962) · Zbl 0112.00603
[68] Mendelsohn, E., Every group is the collineation group of some projective plane, J. of Geometry, 2, 97-106 (1972) · Zbl 0235.50013
[69] Mendelsohn, E., On a technique for representing semigroups as endomorphism semigroups of graphs with given properties, Semigroup Forum, 4, 283-294 (1972) · Zbl 0262.20083
[70] Mendelsohn, E., Pathological projective planes: associate affine planes, J. Geom., 4, 161-165 (1974) · Zbl 0266.50024
[71] Mendelsohn, E., On the groups of automorphisms of Steiner tripel and quadrupel systems, J. Comb. Th., A 25, 97-104 (1978)
[72] Mendelsohn, E., Every (finite) group is the group of automorphisms of a (finite) strongly regular graph, Ars Combinatoria, 6, 75-86 (1978) · Zbl 0455.05035
[73] Moon, J. W., Tournaments with given automorphism group, Canad. J. Math., 16, 485-489 (1964) · Zbl 0121.40204
[74] Ostrom, T.; Wagner, A., On projective and affine planes with transitive collineation groups, Math. Zeitschr., 71, 186-199 (1959) · Zbl 0085.14302
[75] A.Pasini,On the free σ-structures, Preprint. · Zbl 0512.51007
[76] Permutti, R., Una generalizzazione dei piani di Möbius, Le Matematiche, 22, 360-374 (1967) · Zbl 0176.18001
[77] Pickert, G., Projektive Ebenen (1955), Berlin: Springer, Berlin
[78] Sabidussi, G., Graphs with given automorphism group and given graph theoretical properties, Canad. J. Math., 9, 515-525 (1957) · Zbl 0079.39202
[79] Sabidussi, G., Graphs with given infinite group, Monatsh. Math., 64, 446-457 (1960)
[80] Salzmann, H., Homogene kompakte projektive Ebenen, Pac. J. Math., 60, 2, 217-234 (1975) · Zbl 0323.50009
[81] Schafarevitsch, I. R., Construction of fields of algebraic numbers with given solvable Galois group, Izvestiya Akad. Nauk SSSR Ser. Mat., 18, 525-578 (1954) · Zbl 0057.27401
[82] Schleiermacher, A.; Strambach, K., Freie Erweiterungen in der affinen Geometrie und der Geometrie der Kreise (I u. II), Abh. Math. Sem. Univ. Hamburg, 34, 22-37 (196970) · Zbl 0195.22301
[83] Schmidt, E. T., Universelle Algebren mit gegebenen Automorphismengruppen und Unteralgebrenverbänden, Acta Sci. Math. Szeged, 24, 251-254 (1963) · Zbl 0113.24901
[84] E.Schreiber, Freie Strukturen und die Gruppe der affinen Projektivitäten, Dissertation, Erlangen, 1979.
[85] Sichler, J., Group-universal unary varieties, Alg. Universalis, 11, 12-20 (1980) · Zbl 0449.08003
[86] Somma, C.; Barlotti, A., Generalised quadrangles with parallelism, Combinatorial and Geometric Structures and Their Applications, 265-282 (1977), Amsterdam-New York-Oxford: North Holland, Amsterdam-New York-Oxford
[87] K.Strambach,Gruppenuniversalität von Geometrien, in: «2. Kolloquium über diskrete Geometrie», Ergänzungsband, Institut für Mathematik der Universität Salzburg, (1980), pp. 273-290.
[88] Tits, J., Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Inventiones Math., 43, 283-295 (1977) · Zbl 0399.20037
[89] Yasuhara, M., The amalgamation property, the universalhomogeneous models, and the generic models, Math. Scand., 34, 5-36 (1974) · Zbl 0298.02053
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