Embedding theorems for solvable groups

V. Roman’kov
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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 this http URL. 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 $H\in {\mathcal M}{\mathcal A}$.
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可解群的嵌入定理
在本文中,我们证明了在具有少量生成器的群中群嵌入的一系列结果。我们证明了${\mathcal M}$中的每一个有限生成的群$G$可以嵌入$4$生成的群$H ${\mathcal M}{\mathcal a}$ (${\mathcal a}$表示阿贝尔群的种类)。如果$G$是有限群,则$H$也可以被发现是有限群。由此可见,任何导出长度为$l$的有限生成(有限)可解群$G$可以嵌入到长度为$l+1$的$4$生成(有限)可解群$H$中。因此,我们回答了V. H. Mikaelian和这个http URL的问题。Olshanskii。还证明了任意可数群$G\in {\mathcal M}$,使得阿贝尔化$G_{ab}$是一个自由阿贝尔群,可嵌入$2$生成的群$H\in {\mathcal M}{\mathcal a}$。
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