论文标题
嵌入可溶剂组的定理
Embedding theorems for solvable groups
论文作者
论文摘要
在本文中，我们证明了一系列的结果，这些嵌入在具有少量发电机的组中。我们表明，每个有限生成的组$ g $位于品种$ {\ Mathcal m} $中可以嵌入$ 4 $生成的组$ h \ in {\ Mathcal M} {\ Mathcal M} {\ Mathcal a} $（$ {$ {\ MATHCAL A} $表示各种Abelian群体）。如果$ g $是一个有限的组，则$ h $也可以作为有限组找到。随之而来的是，任何有限生成的（有限的）可解决的组$ g $的衍生长度$ l $的$ g $都可以嵌入$ 4 $的（有限的）可解决的（有限的）$ h $长度$ l+1 $中。因此，我们回答了V. H. Mikaelian和A.yu的问题。 Olshanskii。还表明，{\ Mathcal m} $中的任何可数组$ g \ in {\ mathcal m} $，以便abelianization $ g_ {ab {ab} $是一个免费的abelian组，可以嵌入$ 2 $的$ 2 $生成的组$ h \ in {\ Mathcal M} {\ Mathcal a} $中。
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 $H\in {\mathcal M}{\mathcal A}$.
