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On normal verbal embeddings of some classes of groups. (English) Zbl 1094.20009
A normal embedding of the group $$G$$ into the group $$H$$ is verbal for the word set $$V\subseteq F_\infty$$ if the corresponding isomorphic image $$\overline G$$ of $$G$$ lies in the verbal subgroup $$V(H)$$ of $$H$$: $$G\cong\overline G\vartriangleleft H$$, $$\overline G\subseteq V(H)$$.
In Section 2 of the article a short and more effective proof of the following result is given. Theorem 1. Let $$G$$ be an arbitrary group and $$V\subseteq F_\infty$$ be a non-trivial word set. Then there exists a group $$H=H(G,V)$$ with a normal subgroup $$\overline G\subseteq V(H)$$ isomorphic to $$G$$ if and only if $$V(\operatorname{Aut}(G))\supseteq\text{Inn}(G)$$.
In the proof of Theorem 1 a new construction (similar to wreath product) is used. With the use of this construction a criterion for normal verbal embeddings of soluble groups into soluble groups is established (Theorem 2). In Section 3 similar questions for nilpotent groups are considered (Theorems 3, 4, 5). In Section 4 normal verbal embeddings for $$SN^*$$-groups are established. In Section 5 a few normal verbal embedding constructions for the most “well-known” words are considered.

##### MSC:
 20E10 Quasivarieties and varieties of groups 20E07 Subgroup theorems; subgroup growth 20E22 Extensions, wreath products, and other compositions of groups 20F16 Solvable groups, supersolvable groups 20F18 Nilpotent groups
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