Monatshefte für Mathematik

, Volume 144, Issue 4, pp 265–273 | Cite as

Circularity of Finite Groups without Fixed Points

  • Kostia I. Beidar
  • Wen-Fong Ke
  • Hubert Kiechle
Article

Abstract.

Let Φ be a fixed point free group given by the presentation \(\langle A, B\,\vert\, A^\mu=1,\, B^\nu=A^t,\, BAB^{-1}=A^\rho\rangle\) where μ and ρ are relative prime numbers, t = μ/s and s = gcd(ρ − 1,μ), and ν is the order of ρ modulo μ. We prove that if (1) ν = 2, and (2) Φ is embeddable into the multiplicative group of some skew field, then Φ is circular. This means that there is some additive group N on which Φ acts fixed point freely, and |(Φ(a)+b)∩(Φ(c)+d)| ≤ 2 whenever a,b,c,dN, a≠0≠c, are such that Φ(a)+b≠Φ(c)+d.

2000 Mathematics Subject Classifications: 20D60, 51E05 
Key words: Circular, Ferrero pair, metacyclic group, skewfield 

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Copyright information

© Springer-Verlag/Wien 2005

Authors and Affiliations

  • Kostia I. Beidar
    • 1
  • Wen-Fong Ke
    • 1
  • Hubert Kiechle
    • 2
  1. 1.National Cheng Kung UniversityTainanTaiwan
  2. 2.Universität HamburgGermany

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