Abstract
We give three proofs, two intrinsic and one extrinsic, that every Dickson–Ganley unital \({\mathcal{U}(\sigma)}\), parametrized by a field automorphism σ, is non-classical if σ is not the identity, extending a result of Ganley’s (Math Z 128:34–42, 1972); we prove that \({\mathcal{U}(\sigma_1)}\) is isomorphic to \({\mathcal{U}(\sigma_2)}\) if and only if σ 1 = σ 2 or σ 1 = σ −12 ; and we determine the (design) automorphism group of \({\mathcal{U}(\sigma)}\) as the collineation subgroup of the ambient Dickson semifield plane stabilizing the unital. This contains as a special case the corresponding result of O’Nan’s (J Algebra 20:495–511, 1965) on the classical unital.
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This work was partially supported by a grant from the Research Grant Council of the HKSAR, China (Project number: HKU7060/11P)
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Hui, A.M.W., Law, H.F., Tai, Y.K. et al. Non-classical polar unitals in finite Dickson semifield planes. J. Geom. 104, 469–493 (2013). https://doi.org/10.1007/s00022-013-0174-2
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DOI: https://doi.org/10.1007/s00022-013-0174-2