Abstract
We show that conjugacy classes of Baer involutions and non-elliptic polarities, respectively, of proper (i.e., non-desarguesian) Moufang planes are interrelated. Restriction of the conjugating group to the stabilizer of a triangle or a quadrangle does not refine the classes. These results are applied to prove transitivity properties for the centralizers of these polarities. Along the way, a new proof is obtained for the fact that the automorphism group of a Moufang plane acts transitively on quadrangles.
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Communicated by A. Constantin.
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Knarr, N., Stroppel, M. Polarities and planar collineations of Moufang planes. Monatsh Math 169, 383–395 (2013). https://doi.org/10.1007/s00605-012-0409-6
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DOI: https://doi.org/10.1007/s00605-012-0409-6
Keywords
- Moufang plane
- Translation plane
- Baer involution
- Polarity
- Conjugacy
- Semifield
- Division algebra
- Alternative algebra
- Composition algebra
- Octonion field
- Automorphism
- Autotopism