Projective special unitary groups

The projective special unitary group PSU(n,𝔽q) is the special unitary group SU(n,𝔽q) modulo its center.

• Order — |PSU(n,𝔽q)| = |SU(n,𝔽q)| / gcd(n, q + 1) — the center of SU(n,𝔽q) consists of the scalars λI with λq+1 = 1 and λⁿ = 1.
• Simple — for every n ≥ 2 and prime power q except PSU(2,𝔽2) ≅ S₃, PSU(2,𝔽3) ≅ A₄, and PSU(3,𝔽2) (order 72, solvable) — the classical simple groups of Lie type ²An−1.
• Dimension 2 — PSU(2,𝔽q) = PSL(2,𝔽q), inheriting the PSL exceptions.

Small cases

• PSU(2,𝔽5) = PSL(2,𝔽5) ≅ A₅ (order 60).
• PSU(2,𝔽9) = PSL(2,𝔽9) ≅ A₆ (order 360).
• PSU(3,𝔽3) — order 6048, simple; the smallest simple PSU that is not already a PSL.
• PSU(4,𝔽2) — order 25 920, simple; exceptionally isomorphic to PSp(4,𝔽3) (PSp), so one group carries both names.

Part of the wider family of matrix groups.

Wikipedia: Unitary group