# On involutions of type $\mathrm{O}(\mathrm{q},k)$ over a field of characteristic two

## TL;DR

This paper classifies involutions in orthogonal groups over fields of characteristic two, detailing conjugacy classes based on their geometric actions and providing a complete classification for non-defective quadratic spaces.

## Contribution

It offers a comprehensive classification of involutions in orthogonal groups over characteristic two fields, including a complete classification for non-defective cases and insights into defective cases.

## Key findings

- Complete classification of involutions in non-defective quadratic spaces
- Description of involutions as products of transvections and hyperbolic involutions
- Discussion of involutions acting on the radical of the space

## Abstract

In this article we study the involutions of $\mathrm{O}(V,\mathrm{q})$, an orthogonal group for a vector space $V$ with quadratic form $\mathrm{q}$ over a field of characteristic 2. The classification proceeds by discussing conjugacy classes of involutions arising as a product of transvections, involutions with respect to a hyperbolic space, and involutions acting nontrivially in the radical of $V$. We achieve a complete classification of the conjugacy classes of involutions when the quadratic space $(V,\mathrm{q})$ is non-defective, and conclude with a discussion of the defective case.