Embedding theorems for solvable groups

Abstract

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 G lying in a variety M {\mathcal M} can be embedded in a 4 4 -generated group H ∈ M A H \in {\mathcal M}{\mathcal A} ( A {\mathcal A} means the variety of abelian groups). If G G is a finite group, then H H can also be found as a finite group. It follows, that any finitely generated (finite) solvable group G G of the derived length l l can be embedded in a 4 4 -generated (finite) solvable group H H of length l + 1 l+1 . Thus, we answer the question of V. H. Mikaelian and A. Yu. Olshanskii. It is also shown that any countable group G ∈ M G\in {\mathcal M} , such that the abelianization G a b G_{ab} is a free abelian group, is embeddable in a 2 2 -generated group H ∈ M A H\in {\mathcal M}{\mathcal A} .