On normal verbal embeddings of some classes of groups. (Q1878954)

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 \(V(\Aut(G))\supseteq\text{Inn}(G)\). 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.