Embedding theorems for solvable groups

In this paper, we prove a series of results on group embeddings in groups with small number generators. We show that each finitely generated G G lying variety alttext="script upper M"> M encoding="application/x-tex">{\mathcal M} can be embedded alttext="4"> 4 encoding="application/x-tex">4 -generated H element-of script M A"> H ∈ mathvariant="script">A encoding="application/x-tex">H \in {\mathcal M}{\mathcal A} ( encoding="application/x-tex">H also found as group. It follows, any (finite) solvable derived length alttext="l"> l encoding="application/x-tex">l alttext="l plus 1"> + 1 encoding="application/x-tex">l+1 . Thus, answer question V. H. Mikaelian and A. Yu. Olshanskii. shown countable G encoding="application/x-tex">G\in , such abelianization Subscript b"> a b encoding="application/x-tex">G_{ab} free embeddable alttext="2"> 2 encoding="application/x-tex">2 encoding="application/x-tex">H\in