Automorphisms of (Affine) SL(2,q)-Unitals

TL;DR

This paper investigates the automorphism groups of SL(2,q)-unitals, establishing bounds, fixed parallelisms, and symmetries in their affine and closure forms, advancing understanding of their structural automorphisms.

Contribution

It provides a sharp upper bound for automorphism groups of affine SL(2,q)-unitals and identifies fixed parallelisms and blocks under automorphisms, revealing new symmetry properties.

Findings

Sharp upper bound for automorphism groups

Exactly two parallelisms fixed by all automorphisms

Existence of a fixed block under full automorphism group

Abstract

$\operatorname{SL}(2,q)$-unitals are unitals of order $q$ admitting a regular action of $\operatorname{SL}(2,q)$ on the complement of some block. They can be obtained from affine $\operatorname{SL}(2,q)$-unitals via parallelisms. We compute a sharp upper bound for automorphism groups of affine $\operatorname{SL}(2,q)$-unitals and show that exactly two parallelisms are fixed by all automorphisms. In $\operatorname{SL}(2,q)$-unitals obtained as closures of affine $\operatorname{SL}(2,q)$-unitals via those two parallelisms, we show that there is one block fixed under the full automorphism group.