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Coprime commutators in profinite groups
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DOI：10.1007/s13163-026-00573-9.png)
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Abstract
En 中文
By a coprime commutator in a profinite group G we mean any element of the form [x, y], where x,y is an element of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in G$$\end{document} and (|x|,|y|)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(|x|,|y|)=1$$\end{document}. It is well-known that the subgroup generated by the coprime commutators of G is precisely the pronilpotent residual gamma infinity(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _\infty (G)$$\end{document}. There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of gamma infinity(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _\infty (G)$$\end{document} and, more generally, on the structure of G. In this paper we show that if the set of coprime commutators of a profinite group G is covered by countably many procyclic subgroups, then gamma infinity(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _\infty (G)$$\end{document} is finite-by-procyclic. In particular, it follows that G is finite-by-pronilpotent-by-abelian.
Keywords:
Profinite groups
Procyclic groups
Coprime commutators
Journal
R
IF:
1.7
Papers:
34
Citations:
0
