Moufang sets generated by translations in unitals

Abstract

We consider unitals of order q with two points which are centers of translation groups of order q. The group G generated by these translations induces a Moufang set on the block joining the two points. We show that G is either  SL(2, ) q (as in all classical unitals and also in some nonclassical examples), or  PSL(2, )q , or a Suzuki, or a Ree group. Moreover, G is semiregular outside the special block.

Unitals with many involutory translations

“…A translation (with center c) of a unital U is an automorphism of U that fixes the point c and every block through c. We write Γ [c] for the group of all translations with center c. Much is known for the case where there are at least two points u, v such that Γ [u] and Γ [v] both have order q, see [15]. If there even exist three non-collinear points with that property then the unital is the classical hermitian one, see [13].…”