On Embeddings of Countable Generalized Soluble Groups into Two-Generated Groups

Abstract

to professor h. s. mikaelian, my father Strengthening a theorem of L. G. Kovács and B. H. Neumann on embeddings of countable SN * -and SI * -groups into two-generated SN * -and SI * -groups, we establish embeddability of fully ordered countable SN-, SN * -, SI-, and SI * -groups into appropriate fully ordered two-generated groups of the same type. Moreover, for an arbitrary non-trivial word set V ⊆ F ∞ the mentioned two-generated group can be chosen such that the verbal V -subgroup of the latter contains the order isomorphic copy of an initial countable group. These embeddings are subnormal but not, in general, normal.  2002 Elsevier Science (USA)

On a problem on explicit embeddings of the group ℚ

“…(ii) if m/n < m /n in Q, then also Φ(m/n) < Φ(m /n ) in G; (iii) for any g,h,t ∈ G, if g < h, then also gt < ht and tg < th. For further development of this subject, see our recent work [8,9,10]. The embeddings Φ and Ψ both preserve the torsion freeness of Q.…”

Metabelian varieties of groups and wreath products of abelian groups

“…3 and 8.4 in Section 8.1). See also our work [HM00,M00,M02a,M03] for other applications of methods with wreath products.…”

On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$

“…For embedding construction purposes we need to find a fully ordered torsion free nilpotent group S with a non-trivial positive element a ∈ V (S), as it is done in [9]. As a such group we take S = F k (N c ), where c is the least integer, such that N c is not contained in the variety defined by V and k is such that S / ∈ var(F ∞ /V (F ∞ )).…”

“…As a such group we take S = F k (N c ), where c is the least integer, such that N c is not contained in the variety defined by V and k is such that S / ∈ var(F ∞ /V (F ∞ )). A full order relation can be defined in S (see [9], we omit the routine details to much shorten the proof since the exact method of construction of that embedding is immaterial for purposes of this proof).…”