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On finiteness of some verbal subgroups in profinite groups

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arxiv 2009.11775 v2 pith:24IUKWQU submitted 2020-09-24 math.GR

On finiteness of some verbal subgroups in profinite groups

classification math.GR
keywords groupgeneratedgroupsprofinitevaluesverbalwordwords
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Given a group word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. In the present paper we consider profinite groups admitting a word $w$ such that the cardinality of $G_w$ is less than $2^{\aleph_0}$ and $w(G)$ is generated by finitely many $w$-values. For several families of words $w$ we show that under these assumptions $w(G)$ must be finite. Our results are related to the concept of conciseness of group words.

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