$\mathrm{PSU}(n,q)$Projective special unitary finite simple groups

The determinant-one isometries of a nondegenerate Hermitian form over F(q²), modulo scalar matrices.

• Finite simple groups: Lecture 3 — Classical groupsUnitary groups; structure of unitary groups I–IIOpen source

n: integer n ≥ 3; q: q = p^f, with p prime and f ≥ 1

• Finite simple groups: Lecture 3 — Classical groupsUnitary groups; structure of unitary groups I–IIOpen source

Exclude (n,q) = (3,2); q is the fixed-field size, not the matrix-field size.

• Finite simple groups: Lecture 3 — Classical groupsUnitary groups; structure of unitary groups I–IIOpen source

q^[n(n−1)/2] · ∏ᵢ₌₂ⁿ(qⁱ−(−1)ⁱ) / gcd(n,q+1)

• Finite simple groups: Lecture 3 — Classical groupsOrders and scalar center of unitary groupsOpen source

n = 2: PSU(2,q) ≅ PSL(2,q); these are indexed in the linear family.

• Finite simple groups: Lecture 3 — Classical groupsUnitary groups; structure of unitary groups I–IIOpen source

(3,2): PSU(3,2) has order 72 and is not simple.

• Finite simple groups: Lecture 3 — Classical groupsUnitary groups; structure of unitary groups I–IIOpen source

(4,2): PSU(4,2) ≅ PSp(4,3).

• Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic pSection 7, seven exceptional isomorphisms before Proposition 24Open source