Omega groups

Ω(n,𝔽q) — written Ω(n,𝔽q) — is the index-2 subgroup of SO(n,𝔽q) cut out by the spinor norm, and the last step down the orthogonal chain

GO  ⊇  SO  ⊇  Ω

Like GO and SO it carries a Witt type in even dimension: Ω+(2m,𝔽q) and Ω−(2m,𝔽q), with none in odd dimension.

• Order — |Ω| = |GO| / 4 over a field of odd order, and |GO| / 2 over a field of even order, where the determinant is useless (it is identically 1) and the Dickson invariant does the cutting instead.
• Simplicity — Ω itself need not be simple; its projective quotient PΩ = Ω / Z(Ω) is, apart from small exceptions. Z(Ω) is trivial or of order 2.

A naming caution

The ATLAS of Finite Groups writes Onε(q) for the simple group PΩε(n, q) — not for GO, and not for Ω. Three different groups sit behind similar-looking symbols, so a name has to say which one it means. Here Omega±(n, q) is the spinor kernel itself; where it differs from its projective quotient, they differ by a center of order 2.

The exceptional isomorphisms

In dimension at most 6 every orthogonal group coincides with a member of another classical family — the classical exceptional isomorphisms:

So Ω(3,𝔽7) ≅ PSL(2,𝔽7) of order 168, Ω−(4,𝔽3) ≅ PSL(2,𝔽9) ≅ A₆, and Ω+(6,𝔽2) ≅ A₈. In Lie-theoretic terms these are the coincidences B₁ = A₁, B₂ = C₂, D₂ = A₁ × A₁, ²D₂ = ²A₁, D₃ = A₃ and ²D₃ = ²A₃.

References

• Orthogonal group
• Spinor norm
• List of finite simple groups