Bireflectionality in the commutator subgroup of a finite orthogonal group

TL;DR

This paper classifies bireflections and reversible elements in the commutator subgroup of finite orthogonal groups over fields with odd characteristic, revealing specific conditions under which all elements are bireflections.

Contribution

It provides a complete classification of bireflections and reversible elements in the commutator subgroup of finite orthogonal groups, identifying exceptional cases.

Findings

Every reversible element is a bireflection except in specific cases.

All elements are bireflections if they are reversible under certain conditions.

The classification depends on the field characteristic and the dimension of the vector space.

Abstract

We classify the bireflections (products of 2 involutions) in the commutator subgroup G an orthogonal group O(V) over a finite field GF(q) of characteristic not 2. We show that every element of G is a bireflection if it is reversible (conjugate to its inverse in G), except when $q \equiv 3 \mod 4, \dim V \equiv 2 \mod 4$ and $V$ is hyperbolic. We also classify the reversible elements of G.