General unitary groups

The general unitary group GU(n,𝔽q) is the group of all matrices preserving a non-degenerate Hermitian form on 𝔽q²n — the form built from the field automorphism x ↦ xq of 𝔽q² over 𝔽q.

• The parameter is the smaller field. GU(n,𝔽q) is written over 𝔽q, but its matrices have entries in the quadratic extension 𝔽q².
• Order — |GU(n,𝔽q)| = qn(n−1)/2 · ∏i=1n (qi − (−1)i).
• Determinant — a unitary matrix has det · detq = 1, so the determinant lands in the norm-one scalars Cq+1. It maps onto them, with kernel SU, so GU(n,𝔽q) / SU(n,𝔽q) ≅ Cq+1.
• Center — all norm-one scalars λI (λq+1 = 1), cyclic of order q + 1.

The form is unique

Over a finite field, any two non-degenerate Hermitian forms of the same dimension are equivalent — unlike quadratic forms, which split into a + type and a − type in even dimension. So GU(n,𝔽q) is a single group up to isomorphism, with no type superscript.

Small cases

• GU(2,𝔽q) ≅ SL(2,𝔽q) ⋊ Cq+1 — order q(q − 1)(q + 1)². |GU(2,𝔽2)| = 18, |GU(2,𝔽3)| = 96, |GU(2,𝔽4)| = 300.
• |GU(3,𝔽2)| = 648 and |GU(3,𝔽3)| = 24 192. The latter contains SU(3,𝔽3) of order 6048, whose projective quotient is a simple group.

References

• Unitary group
• Classical group