Automorphisms of Unitals

Markus Stroppel, Hendrik van Maldeghem
2006 Bulletin of the Belgian Mathematical Society Simon Stevin  
It is shown that every automorphism of classical unitals over certain (not necessarily commutative) fields is induced by a semi-similitude of a corresponding hermitian form. In particular, this is true if the form uses an involution of the second kind. In [16], J. Tits has studied classical unitals, defined by suitable polarities. Using the Borel-Tits Theorem [2], he determines the full group of automorphisms in the special case where the ground field is commutative and infinite. The finite
more » ... has been treated by M.E. O'Nan [12] . In [16] , it is also claimed that the result is extendable (presumably, within the limitations imposed by the machinery used in [2]: for instance, one would at least require that the ground field has finite dimension over its center). However, no precise statement has been published up to now. In a recent investigation [13] into non-classical unitals in translation planes obtained via modification of the projective plane over Hamilton's quaternions, complete information about automorphisms of unitals over the quaternions is required, in order to distinguish the unitals in question from the classical one. We generalize the results of [16], using "elementary methods" in the sense of Dieudonné's review [6]: we treat the groups in question as classical groups rather than as algebraic groups. For many ground fields, we show that a distinguished subgroup of the automorphism group of the unital contains all unitary reflections, and that the set of reflections is invariant in the group of all automorphisms of the unital. See Sections 3 and 5. The reflections are then used (in Section 6) to reconstruct the ambient projective plane, leading to a determination of the full group of automorphisms under some technical assumptions. For instance, this is possible in the cases where an involution 2000 Mathematics Subject Classification : primary 51A10, secondary 11E57, 14L35, 51N30, 51A50, 51A45,
doi:10.36045/bbms/1136902624 fatcat:arncudf3fjbrvlramxu72eanoa