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Multiplicative loops of topological quasifields

Volume 113 / 2017

Ágota Figula Banach Center Publications 113 (2017), 123-134 MSC: 20N05; Secondary 22A30, 12K10, 51A40, 57M60. DOI: 10.4064/bc113-0-8


Locally compact connected topological non-Desarguesian translation planes have been a popular subject for research in geometry since the seventies of the last century. These planes are coordinatized by locally compact quasifields $(Q,+, \cdot )$ such that the kernel of $Q$ is either the field $\mathbb R$ of real numbers or the field $\mathbb C$ of complex numbers. In recent papers we determined the algebraic structure of the multiplicative loops $Q^{\ast }=(Q \setminus \{ 0 \}, \cdot )$ of quasifields $Q$ such that $Q$ has dimension $2$ over its kernel. Now we compare these cases and give a unified treatment of our results. In particular, we deal with multiplicative loops which either have a one-dimensional normal subloop or contain a compact subgroup.


  • Ágota FigulaInstitute of Mathematics
    University of Debrecen
    P.O.Box 400
    H-4002, Debrecen, Hungary

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