Bireflectionality in the commutator subgroup of a finite orthogonal group

2024/11/02 by Nielsen, Klaus
#15A15 #15F10 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2411.01323

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 ≡ 3 \mod 4, dim V ≡ 2 \mod 4 and V is hyperbolic. We also classify the reversible elements of G.

Related

• Bireflections of the commutator subgroup of an orthogonal group over the reals
• Bireflectionality in special orthogonal groups
• Products of two involutions in orthogonal and symplectic groups
• A note on bireflectional elements of an algebra
• Products of involutions in symplectic groups I: bireflections