  In this paper, we prove a series of results on group embeddings in groups
with a small number of generators. We show that each finitely generated group
$G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group
$H \in {\mathcal M}{\mathcal A}$ (${\mathcal A}$ means the variety of abelian
groups). If $G$ is a finite group, then $H$ can also be found as a finite
group. It follows, that any finitely generated (finite) solvable group $G$ of
the derived length $l$ can be embedded in a $4$-generated (finite) solvable
group $H$ of length $l+1$. Thus, we answer the question of V. H. Mikaelian and
A.Yu. Olshanskii. It is also shown that any countable group $G\in {\mathcal
M}$, such that the abelianization $G_{ab}$ is a free abelian group, is
embeddable in a $2$-generated group $H\in {\mathcal M}{\mathcal A}$.
