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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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