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Results in Mathematics

, Volume 11, Issue 3–4, pp 272–288 | Cite as

Normalteiler vom Geschlecht eins in freien Produkten endlicher zyklischer Gruppen

  • Gabriele Kern-Isberner
  • Gerhard Rosenberger
Article
  • 7 Downloads

Abstract

M. Newman [8] gave a complete group-theoretic description of the normal subgroups of genus one of the modular group PSL(2,Z) ≅ Z2*Z3. In this paper we generalize his result and give a characterization of the normal subgroups of genus one of free products of finitely many finite cyclic groups. In particular we give a complete group-theoretic description of the normal subgroups of genus one of the next important Hecke-group \({\rm G}(\sqrt 2)\cong z_{2}\ast z_{4}\). As an interesting corollary we get the following. If b_(n) is the number of normal subgroups of \({\rm G({\sqrt 2})}\) of index kn and genus 1 (n ∈ N) then \({\rm b_{1}(n)={1\over 4}\ \#\ \lbrace (x,y)\in Z^{2}\mid x^{2}+y^{2}=n\rbrace}.\)

Keywords

Normal Subgroup Free Product Fuchsian Group Finite Cyclic Group 
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Copyright information

© Birkhäuser Verlag, Basel 1987

Authors and Affiliations

  • Gabriele Kern-Isberner
    • 1
  • Gerhard Rosenberger
    • 2
  1. 1.Krefeld 11Bundesrepublik Deutschland
  2. 2.Fachbereich MathematikUniversität DortmundDortmund 50Bundesrepublik Deutschland

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