Regular automorphism groups on partial geometries |
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| Institution: | 1. Laboratory of Integrative Neuroscience (LiNC), Federal University of Sao Paulo (UNIFESP), Brazil;2. Early Psychosis Group (GAPi), Federal University of Sao Paulo (UNIFESP), Brazil;3. Genetics Division, Department of Morphology and Genetics, Universidade Federal de São Paulo (UNIFESP), São Paulo, Brazil;4. Department of Psychiatry, Pontificia Universidad Católica de Chile, Santiago, Chile;5. Early Intervention Program, Instituto Psiquiátrico Dr J. Horwitz Barak, Santiago, Chile;6. Department of Neurology and Psychiatry, Faculty of Medicine, Clínica Alemana Universidad del Desarrollo, Santiago, Chile;7. Universidad de Antioquia, Medellin, Colombia |
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| Abstract: | Recently, Ghinelli (Geom Dedicata (1992) 165–174) had studied the generalized quadrangles which admits automorphism groups acting regularly on the points. In this paper, we generalize her idea to partial geometries, pg(s + 1, t + 1, α). Some examples and basic properties are given. In particular, we prove that under certain conditions on the automorphism group and the lines, such a geometry is a translation net. Applying the results to the case when s = t and the automorphism group G is abelian, we find that either the geometry is a translation net or all the lines of the geometry are generated by a subset of G. Also, for this case, we conjecture that the parameter α is either s or s + 1, except (s, α) = (5, 2), and we have checked that it is true for s ? 500. |
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