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On the Group of Continuous Automorphisms of Some Profinite Groups

Authors
  • Mantika, Gilbert1
  • Tieudjo, Daniel2
  • 1 The University of Maroua, Department of Mathematics, Ecole Normale Supérieure, Maroua, Cameroon , Maroua (Cameroon)
  • 2 The University of Ngaoundere, Department of Mathematics and Computer Science, Ngaoundere, Cameroon , Ngaoundere (Cameroon)
Type
Published Article
Journal
Lobachevskii Journal of Mathematics
Publisher
Pleiades Publishing
Publication Date
Mar 21, 2018
Volume
39
Issue
2
Pages
243–251
Identifiers
DOI: 10.1134/S1995080218020191
Source
Springer Nature
Keywords
License
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Abstract

We prove some conditions on a given abstract group G, such that the group Autc(G^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat G$$\end{document}) of the continuous automorphisms of the profinite completion G^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat G$$\end{document} of G endowed with the congruence subgroup topology, is profinite. Also, for a given abstract group G, if Autc(G^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat G$$\end{document}) is profinite, then we establish relations betweenG, Aut(G), Aut(G)^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {Aut(G)}$$\end{document}, and Autc(G^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat G$$\end{document}) when each of these groups is endowed with appropriate topology. Finally, we applied the obtained results to the class of one-relator groups given by the presentation Gmn = 〈a, b; [am, bn] = 1〉 (m > 1,n > 1).

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