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An Embedding Construction for Ordered Groups

To Professor Alexander Yurievich Ol'shanskii, my teacher

Part of: Special aspects of infinite or finite groups Structure and classification 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: [email protected]
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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

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