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On normal verbal embeddings of groups. (English) Zbl 0953.20017
Let \(H\) be a group and \(V\) a set of words. An \(nV\)-embedding \(\nu\) of \(H\) into a group \(G\) is a monomorphism \(\nu\colon H\mapsto G\) such that \(\nu(H)\) is a normal subgroup of \(G\) and \(\nu(H)\subseteq V(G)\). The group \(H\) is said to be \(nV\)-embeddable if there is a group \(G\) and an \(nV\)-embedding of \(H\) into \(G\).
The main result of the article (Theorem 1) is the following: if \(V\) is a set of words and \(H\) is a group then \(H\) is \(nV\)-embeddable if and only if \(V(\operatorname{Aut}(H))\supseteq\text{Inn}(H)\). Theorem 2 states that if \(H\) is a non-trivial Abelian group then \(H\) is \(nV\)-embeddable into a nilpotent group \(G\) and that \(H\) is \(nV\)-embeddable into an Abelian group if and only if \(V\) has some consequence of the form \(x^n=1\).
Theorem 3 deals with the case when \(H\) is solvable and \(V\) consists of one of the words \([x_1,x_2]\), \(x^\ell\), \(\gamma_c(x_1,\dots,x_c)\) and \(\delta_n(x_1,\dots,x_{2^n})\) where \(\gamma_c(x_1,\dots,x_c)=[x_1,\dots,x_c]\) and \(\delta_n(x_1,\dots,x_{2^n})\) is defined as follows: \(\delta_0=x\) and \(\delta_{n+1}(x_1,\dots,x_{2^{n+1}})=[\delta_n(x_1,\dots,x_{2^n}),\delta_n(x_1,\dots,x_{2^n})]\). In this case \(H\) is \(nV\)-embeddable into some solvable group \(G\) if and only if \(\operatorname{Aut}(H)\) contains a solvable subgroup \(B\) with \(V(B)\supseteq\text{Inn}(H)\). Theorem 4 states that if \(V\) consists of the word \([x_1,x_2]\) then the nilpotent group \(H\) is \(nV\)-embeddable into some nilpotent group \(G\) if and only if there is a nilpotent subgroup \(B\) of \(\operatorname{Aut}(H)\) such that \(V(B)\supseteq\text{Inn}(H)\) and the extension of \(H\) by \(B\) is nilpotent.
The last result (Theorem 5) is devoted to \(nV\)-embeddings of symmetric groups. It states that (i) the symmetric groups \(S_2\) and \(S_1\) and symmetric groups over an infinite set are \(nV\)-embeddable; (ii) the groups \(S_n\) with \(n\geq 3\), \(n\neq 6\) are \(nV\)-embeddable if and only if \(S_n=V(S_n)\); (iii) the group \(S_6\) is \(nV\)-embeddable if and only if \(\text{Inn}(S_6)\subseteq V\langle\text{Inn}(S_6),\omega\rangle\) where \(\omega\) is an arbitrary outer automorphism of \(S_6\).

MSC:
20E10 Quasivarieties and varieties of groups
20E07 Subgroup theorems; subgroup growth
20E22 Extensions, wreath products, and other compositions of groups
20F28 Automorphism groups of groups
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