On normal verbal embeddings of groups
type: Item
instance of: scholarly article
title: On normal verbal embeddings of groups
zbMATH Open document ID: 0953.20017
DOI: 10.1007/BF02677503
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-09-14T00:00:00Z
review text: 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 \(V(\Aut(H))\supseteq\text{Inn}(H)\). 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 \(\Aut(H)\) 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 \(\Aut(H)\) 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 grou…
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 1506087
zbMATH Keywords: groups
zbMATH Keywords: verbal subgroups
zbMATH Keywords: normal subgroups
zbMATH Keywords: embeddings
zbMATH Keywords: Abelian groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: symmetric groups
author: Hermann Heineken
author: Vahagn H. Mikaelian
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: Q3241238
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description: scientific article; zbMATH DE number 1506087