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On embedding properties of \(SD\)-groups. (English) Zbl 1078.20033
Since the seminal paper of G. Higman, B. H. Neumann and H. Neumann [J. Lond. Math. Soc. 24, 247-254 (1950; Zbl 0034.30101)] showing that any countable group can be embedded in a two-generator group, the problem of embedding various types of countable groups in groups with appropriate properties has been of interest. The groups considered in this paper are \(SD\)-groups, that is, groups in which the commutator series \(G=G^{(0)}\geq G^{(1)}\geq G^{(2)}\geq\cdots\geq G^{(\sigma)}\geq\cdots\) reaches 1 for some finite or infinite ordinal \(\rho\).
The main theorem shows that not only can every countable \(SD\)-group \(G\) be embedded as a subnormal subgroup of a two generator \(SD\)-group \(H\), but the embedding can be chosen so that the image of \(G\) lies in any specified verbal subgroup of \(H\). Moreover, if \(G\) is ordered then \(H\) can be chosen to be ordered and the image of \(G\) to be order-isomorphic to \(G\), and if \(G\) is torsion free, then \(H\) can be chosen to be torsion-free.
The verbal embedding property is then used to show that there exists a continuum of torsion-free, not locally soluble two-generator \(SD\)-groups which generate pairwise distinct varieties of groups.

MSC:
20F14 Derived series, central series, and generalizations for groups
20F19 Generalizations of solvable and nilpotent groups
20E10 Quasivarieties and varieties of groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F60 Ordered groups (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
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