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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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