Special unitary groups

The special unitary group SU(n,𝔽q) is the group of determinant-1 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. SU(n,𝔽q) is written over 𝔽q, but its matrices have entries in the quadratic extension 𝔽q².
• Order — |SU(n,𝔽q)| = |GU(n,𝔽q)| / (q + 1), where |GU(n,𝔽q)| = qn(n−1)/2 · ∏i=1n (qi − (−1)i). The determinant maps GU(n,𝔽q) onto the norm-one scalars Cq+1 with kernel SU(n,𝔽q).
• Center — the scalars λI with λq+1 = 1 and λⁿ = 1, cyclic of order gcd(n, q + 1); the quotient is PSU.

Dimension 2

SU(2,𝔽q) ≅ SL(2,𝔽q) for every prime power q — the unitary and special linear groups coincide in rank 1.

• SU(2,𝔽2) ≅ S₃ (order 6).
• SU(2,𝔽3) ≅ SL(2,𝔽3) (order 24), the binary tetrahedral group.
• SU(2,𝔽5) ≅ SL(2,𝔽5) (order 120), the binary icosahedral group.

Beyond dimension 2

• SU(3,𝔽2) — order 216, solvable.
• SU(3,𝔽3) — order 6048 and simple: gcd(3, 4) = 1, so the center is trivial and SU(3,𝔽3) = PSU(3,𝔽3).
• SU(4,𝔽2) — order 25 920, simple; gcd(4, 3) = 1 again makes it equal to PSU(4,𝔽2), which is exceptionally isomorphic to PSp(4,𝔽3) (PSp).

Part of the wider family of matrix groups.

Wikipedia: Unitary group