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An embedding construction for ordered groups. (English) Zbl 1060.20024
The following main theorem is proved: Let $$G$$ be a countable group and $$V$$ be a nontrivial word set. There is a two-generator group $$H$$ with subgroup $$L$$ such that (i) $$G$$ is isomorphic to $$L$$ and (ii) $$L$$ is a subnormal in $$H$$ and contained in $$V(H)$$. Moreover, $$H$$ can be chosen to be (a) soluble, (b) fully ordered, (c) torsion-free, whenever $$G$$ has the respective property. Subnormality can not be strengthened to normality. – This strengthens and generalizes results of G. Higman, B. H. and H. Neumann, and Dark.

##### MSC:
 20E15 Chains and lattices of subgroups, subnormal subgroups 20E22 Extensions, wreath products, and other compositions of groups 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects) 20E10 Quasivarieties and varieties of groups 20E07 Subgroup theorems; subgroup growth 20F05 Generators, relations, and presentations of groups 20F16 Solvable groups, supersolvable groups
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