Unitary group (awaiting review)

The (general) unitary group $\GU(n, q)$ is the group of $n \times n$ unitary matrices with entries in the field $\F_{q^2}$ (the finite field with $q^2$ elements). Here, a matrix is unitary if the inverse of $M$ equals the conjugate transpose of $M$.

For matrices $M$ over the finite field $\F_{q^2}$, the conjugate of $M$ is defined as the matrix obtained by applying the map $x \mapsto x^q$ (i.e. the $r$-th power of the Frobenius automorphism, where $q = p^r$) to each entry of $M$.

The projective (general) unitary group $\PGU(n, q)$ is the quotient of $\GU(n, q)$ by the scalar matrices in $\GU(n, q)$ (equivalently, $\PGU(n, q)$ is the quotient of $\GU(n, q)$ by its center).

The special unitary group $\SU(n, q)$ is the subgroup of $\GU(n, q)$ consisting of those matrices with determinant 1.

The projective special unitary group $\PSU(n, q)$ is the quotient of $\SU(n, q)$ by the scalar matrices in $\SU(n, q)$ (equivalently, $\PSU(n, q)$ is the quotient of $\SU(n, q)$ by its center).

(see [MR:0827219], page x)