Projective special unitary group

Definition
Suppose $$L$$ is a separable quadratic extension of a field $$K$$ and $$\sigma:L \to L$$ is the unique automorphism of $$L$$ that fixes $$K$$ pointwise. The projective special unitary group of degree $$n$$ for this quadratic extension, denoted $$PSU(n,L)$$ (if the extension being referred to is understood), is defined as the quotient of the special unitary group $$SU(n,L)$$ by its center, which is precisely the set of scalar matrices in $$SU(n,L)$$.

For the real and complex numbers
The most typical usage of the term special unitary group is in the context where $$K$$ is the field of real numbers, $$L$$ is the field of complex numbers, and the automorphism $$\sigma$$ is complex conjugation. In this case, the group $$PSU(n,\mathbb{C})$$ is the quotient of $$SU(n,\mathbb{C})$$ by its center, which is a cyclic group of order $$n$$ comprising those scalar matrices whose scalar entry is a $$n^{th}$$ root of unity. The group $$PSU(n,\mathbb{C})$$ is sometimes also denoted $$PSU(n)$$.

For a finite field
If $$K$$ is the (unique up to isomorphism) finite field of size a prime power $$q$$, there is a unique quadratic extension $$L$$ of $$K$$, and this extension is separable. The extension field is the finite field (unique up to isomorphism) of order $$q^2$$. The automorphism $$\sigma$$ is the map $$x \mapsto x^q$$. The special unitary group for this extension may be denoted $$PSU(n,q)$$ (the more standard choice) or $$PSU(n,q^2)$$ (a less standard choice).