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An Embedding Construction for Ordered Groups

To Professor Alexander Yurievich Ol'shanskii, my teacher

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Ordered structures

Published online by Cambridge University Press:  09 April 2009

Vahagn H. Mikaelian
Vahagn H. Mikaelian
Affiliation:
Department of Informatics and Computer Science Yerevan State University 375025 Yerevan Armenia e-mail: mikaelian@e-math.ams.org
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Abstract

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Generalizing and strengthening some well-known results of Higman, B. Neumann, Hanna Neumann and Dark on embeddings into two-generator groups, we introduce a construction of subnormal verbal embedding of an arbitrary (soluble, fully ordered or torsion free) ordered countable group into a twogenerator ordered group with these properties. Further, we establish subnormal verbal embedding of defect two of an arbitrary (soluble, fully ordered or torsion free) ordered group G into a group with these properties and of the same cardinality as G, and show in connection with a problem of Heineken that the defect of such an embedding cannot be made smaller, that is, such verbal embeddings of ordered groups cannot in general be normal.

Keywords

embeddings of groupsordered groupstwo-generator groups.

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups 20E15: Chains and lattices of subgroups, subnormal subgroups 20E22: Extensions, wreath products, and other compositions 20F14: Derived series, central series, and generalizations 06F15: Ordered groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 3 , June 2003 , pp. 379 - 392
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Dark, R., ‘On subnormal embedding theorems of groups’, J. London Math. Soc. 43 (1968), 387–390.CrossRefGoogle Scholar
[2]Fuchs, L., Partially ordered algebraic systems (Pergamon Press, Oxford, London, 1963).Google Scholar
[3]Hall, P., ‘The frattini subgroups of finitely generated groups’, Proc. London Math. Soc. (3) 11 (1961), 327–352.CrossRefGoogle Scholar
[4]Hall, P., ‘On the embedding of a group into a join of given groups’, J. Austral. Math.Soc. 17 (1974), 434–495.CrossRefGoogle Scholar
[5]Heineken, H., ‘Normal embeddings of p-groups into p-groups’, Proc. Edinburgh Math. Soc. 35 (1992), 309–314.CrossRefGoogle Scholar
[6]Heineken, H. and Mikaelian, V. H., ‘On normal verbal embeddings of groups’, J. Math. Sci., New York 100 (2000), 1915–1924.CrossRefGoogle Scholar
[7]Higman, G., Neumann, B. and Neumann, Hanna, ‘Embedding theorems for groups’, J. London Math. Soc. (3) 24 (1949), 247–254.CrossRefGoogle Scholar
[8]Kaplanski, I., Infinite abelian groups (Univ. Michigan Publ., Ann Arbor, MI, 1954).Google Scholar
[9]Kargapolov, M. I. and Merzlyakov, Ju. I., Fundamentals of the theory of groups, 3rd Edition, (Russian); English translation of the 2nd Edition by Burns, R. G. (Springer, New York, 1979) (Nauka, Moscow, 1982).CrossRefGoogle Scholar
[10]Kovács, L. G. and Neumann, B. H., ‘An embedding theorem for some countable groups’, Acta Sci. Math. (Szeged) 26 (1965), 139–142.Google Scholar
[11]Levi, F. W., ‘Ordered groups’, Proc. Indian Acad. Sci. 16 (1942), 256–263.CrossRefGoogle Scholar
[12]Levi, F. W., ‘Contributions to the theory of ordered groups’, Proc. Indian Acad. Sci. 17 (1943), 199–201.CrossRefGoogle Scholar
[13]Levin, F. and Rosenberger, G., ‘A class of SQ-universal groups’, in: Group theory, Singapore, 1987 (de Gruyter, Berlin, 1989) pp. 409–415.Google Scholar
[14]Mikaelian, V. H., ‘Embedding of countable groups into 2-generator groups with additional properties’, in: Abstracts of the International Congress of Mathematicians, Berlin (1998) pp. 31.Google Scholar
[15]Mikaelian, V. H., ‘Subnormal embedding theorems for groups’, J. London Math. Soc. 62 (2000), 398–406.CrossRefGoogle Scholar
[16]Mikaelian, V. H., ‘On varieties of groups generated by wreath products of abelian groups’, in: Abelian groups, rings and modules (Perth 2000), Contemp. Math. 273 (Amer. Math. Soc., Providence, RI, 2001) pp. 223–238.CrossRefGoogle Scholar
[17]Mikaelian, V. H., ‘On embeddings of countable generalized soluable groups in two-generated groups’, J. Algebra 250 (2002), 1–17.CrossRefGoogle Scholar
[18]Mikaelian, V. H., ‘Über die normalen Einbettungen der geordneten Gruppen’, (in preparation) (German).Google Scholar
[19]Neumann, B. H., ‘On ordered groups’, Amer. J. Math. 71 (1949), 1–18.CrossRefGoogle Scholar
[20]Neumann, B. H., ‘Embedding theorems for ordered groups’, J. London Math. Soc. 35 (1960), 503–512.CrossRefGoogle Scholar
[21]Neumann, B. H., ‘Embedding theorems for groups’, Nieuw Arch. Wisk. (3) 16 (1968), 73–78.Google Scholar
[22]Neumann, B. H., and Neumann, Hanna, ‘Embedding theorems for groups’, J. London Math. Soc. 34 (1959), 465–479.CrossRefGoogle Scholar
[23]Neumann, Hanna, Varieties of groups (Springer, Berlin, 1968).Google Scholar
[24]Neumann, P. M., ‘On the structure of standard wreath products of groups’, Math. Z. 84 (1964), 343–373.CrossRefGoogle Scholar
[25]Ol'shanskii, A. Yu., ‘Efficient embeddings of countable groups’, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 105 (1989), 28–34. (Russian).Google Scholar
[26]Ol'shanskii, A. Yu., ‘Embedding of countable periodic groups in simple 2-generator periodic groups’, Ukrain. Mat. Zh. 43 (1991), 980–986. (Russian); English translation: Ukrainian Math. J. 43 (1992), 914–919.Google Scholar
[27]Robinson, D. J. S., A course in the theory of groups, 2nd Edition (Springer, New York, 1996).CrossRefGoogle Scholar
[28]Wilson, J. S. and Zalesskii, P. A., ‘An embedding theorem for certain residually finite groups’, Arch. Math. (Basel) 67 (1996), 177–182.CrossRefGoogle Scholar