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Lobachevskii Journal of Mathematics

, Volume 39, Issue 2, pp 243–251 | Cite as

On the Group of Continuous Automorphisms of Some Profinite Groups

  • Gilbert Mantika
  • Daniel Tieudjo
Article

Abstract

We prove some conditions on a given abstract group G, such that the group Aut c (\(\hat G\)) of the continuous automorphisms of the profinite completion \(\hat G\) of G endowed with the congruence subgroup topology, is profinite. Also, for a given abstract group G, if Aut c (\(\hat G\)) is profinite, then we establish relations betweenG, Aut(G), \(\widehat {Aut(G)}\), and Aut c (\(\hat G\)) 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 G mn = 〈a, b; [a m , b n ] = 1〉 (m > 1,n > 1).

Keywords and phrases

Profinite groups profinite completions profinite topology congruence subgroup topology 

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References

  1. 1.
    G. Arzhantseva, P. de la Harpe, D. Kahrobaei, and Z. Sunic, “The true prosoluble completion of a group: Examples and open problems,” Geom. Dedicata 124, 5–26 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G. Baumslag, “On the residual finiteness of generalised free products of nilpotent groups,” Trans. Am.Math. Soc. 106, 193–209 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    N. Bourbaki, General Topology (Springer, Berlin, 1989).CrossRefzbMATHGoogle Scholar
  4. 4.
    B. Deschamps, Groupes profinis et theorie de Galois (Clarendon, Oxford, 1998) [in French].Google Scholar
  5. 5.
    T. Coulbois, “Propriétés de Ribes-Zalesskii, topologie profinie, produit libre et généralisations,” PhD Thesis (Univ. Paris VII, Paris, 2000).Google Scholar
  6. 6.
    M. Hall, “A topology for free groups and related groups,” Ann.Math. 52, 127–139 (1950).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. D. Loginova, “Residual finiteness of the free product of two groups with commuting subgroups,” Sib. Math. J. 40, 341–350 (1999).MathSciNetzbMATHGoogle Scholar
  8. 8.
    W. Magnus, A. Karass, and D. Solitar, Combinatorial Group Theory (Wiley, New York, London, Sydney, 1966).Google Scholar
  9. 9.
    N. Nikolov and D. Segal, “Finite index subgroups in profinite groups,” C. R. Acad. Sci. Paris 337 (Ser. 1), 303–308 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Puusemp, “Endomorphisms and endomorphisms semi groups of groups,” in Focus on Group Theory Research, Ed. by L.M. Ying (Nova Science, New York, 2006).Google Scholar
  11. 11.
    L. Ribes and P. A. Zalesskii, Profinite Groups (Springer, Berlin, 2010).CrossRefzbMATHGoogle Scholar
  12. 12.
    D. Tieudjo and D. I. Moldavanskii, “Endomorphisms of the group G mn = 〈a, b; [a m, b n] = 1〉, m, n > 1,” J. AfricanMath. Union 9 (3), 11–18 (1998).Google Scholar
  13. 13.
    D. Tieudjo and D. I. Moldavanskii, “On the automorphisms of some one-relator groups,” Commun. Algebra 34, 3975–3983 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. Tieudjo and D. I. Moldavanskii, “Conjugacy separability of some one-relator groups,” Int. J.Math.Math. Sci. 2010, 803243 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. S. Wilson, Profinite Groups (Clarendon, Oxford, 1998).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Ecole Normale SupérieureThe University of MarouaMarouaCameroon
  2. 2.Department of Mathematics and Computer ScienceThe University of NgaoundereNgaoundereCameroon

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