Embedding theorems for solvable groups

TL;DR

This paper proves that finitely generated solvable groups can be embedded into small generating groups with controlled properties, providing new insights into group embeddings and answering longstanding questions.

Contribution

It establishes that finitely generated solvable groups can be embedded into 4-generated solvable groups, and countable groups with free abelianization into 2-generated groups, advancing embedding theory.

Findings

Finitely generated groups in a variety can be embedded into 4-generated groups within the same variety.

Finite solvable groups can be embedded into finite 4-generated solvable groups of derived length increased by one.

Countable groups with free abelianization can be embedded into 2-generated groups within the same variety.

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 $G$ 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 $H\in {\mathcal M}{\mathcal A}$.