Fixed point groups of involutions of type $\mathrm{O}(\mathrm{q},k)$ over a field of characteristic two

TL;DR

This paper classifies the fixed point groups of involutions in orthogonal groups over fields of characteristic two, providing a comprehensive understanding of their structure in different quadratic space cases.

Contribution

It offers a complete classification of fixed point groups of involutions in orthogonal groups over characteristic two fields, covering both totally singular and nonsingular quadratic spaces.

Findings

Full classification of fixed point groups for totally singular quadratic spaces.

Full classification of fixed point groups for nonsingular quadratic spaces.

Implications for general quadratic spaces over characteristic two fields.

Abstract

For $\mathrm{O}(\mathrm{q},k)$, the orthogonal group over a field $k$ of characteristic 2 with respect to a quadratic form $\mathrm{q}$, we discuss the isomorphism classes of fixed points of involutions. When the quadratic space is either totally singular or nonsingular, a full classification of the isomorphism classes is given. We also give some implications of these results for a general quadratic space over a field of characteristic $2$.