A unital in the generalized hexagon of order two, and an exceptional isomorphism between finite groups of Lie type

TL;DR

This paper constructs a model of a Hermitian unital within the octonion algebra over a finite field, revealing an isomorphism between a finite group of Lie type and a group of semi-similitudes, enhancing understanding of their algebraic structures.

Contribution

It provides a novel geometric construction linking the Hermitian unital to octonion algebra, elucidating an exceptional isomorphism between finite Lie type groups.

Findings

Constructed a Hermitian unital model inside octonion algebra over finite field

Demonstrated invariance under automorphisms of the octonion algebra

Explained the isomorphism between finite Lie type group and semi-similitudes group

Abstract

We construct a model of the Hermitian unital of order 3 (obtained from the non-degenerate hermitian form in three variables over the field of order 9) inside the octonion algebra over the field of order 2. This construction is invariant under the automorphism group of that algebra, and explains the known isomorphism from the finite group of exceptional Lie type onto the group of semi-similitudes of the hermitian form.