Projective Unitary Group - Finite Fields

Finite Fields

One can also define unitary groups over finite fields: given a field of order q, there is a non-degenerate Hermitian structure on vector spaces over, unique up to unitary congruence, and correspondingly a matrix group denoted U(n, q) or, and likewise special and projective unitary groups. For convenience, this article with use the convention.

Recall that the group of units of a finite field is cyclic, so the group of units of, and thus the group of invertible scalar matrices in, is the cyclic group of order . The center of has order q+1 and consists of the scalar matrices which are unitary, that is those matrices with . The center of the special unitary group has order and consists of those unitary scalars which also have order dividing n.

The quotient of the unitary group by its center is the projective unitary group, and the quotient of the special unitary group by its center is the projective special unitary group . In most cases ( and ), is a perfect group and is a finite simple group, (Grove 2002, Thm. 11.22 and 11.26).