Special orthogonal groups

The special orthogonal group SO(n,𝔽q) is the determinant-1 subgroup of GO(n,𝔽q). Like GO, it carries a Witt type in even dimension — SO+(2m,𝔽q) and SO−(2m,𝔽q) — and none in odd dimension, where the quadratic form is unique up to scaling.

• Order — |SO| = |GO| / 2 over a field of odd order.
• Even characteristic — every isometry of a quadratic form over 𝔽2k has determinant 1, so SO = GO there and the two names denote the same group.
• Below it — a further index-2 subgroup Omega, cut out by the spinor norm (by the Dickson invariant in even characteristic).

Small cases

• SO(3,𝔽q) ≅ PGL(2,𝔽q) for odd q — the conjugation action of PGL(2,𝔽q) on the trace-zero 2 × 2 matrices preserves the determinant as a quadratic form. So SO(3,𝔽3) ≅ S₄ (order 24) and SO(3,𝔽5) ≅ PGL(2,𝔽5) ≅ S₅.
• SO−(4,𝔽2) = GO−(4,𝔽2) ≅ S₅ and SO+(6,𝔽2) = GO+(6,𝔽2) ≅ S₈ — even characteristic, so these coincide with their GO.

References

• Orthogonal group
• Classical group