Editing Unitary group (section)

=== Finite fields ===
Over the [[finite field]] with {{nowrap|1=''q'' = ''p''<sup>''r''</sup>}} elements, '''F'''<sub>''q''</sub>, there is a unique quadratic extension field, '''F'''<sub>''q''<sup>2</sup></sub>, with order 2 automorphism <math>\alpha\colon x \mapsto x^q</math> (the ''r''th power of the [[Frobenius automorphism]]). This allows one to define a Hermitian form on an '''F'''<sub>''q''<sup>2</sup></sub> vector space ''V'', as an '''F'''<sub>''q''</sub>-bilinear map <math>\Psi\colon V \times V \to K</math> such that <math>\Psi(w, v) = \alpha \left(\Psi(v, w)\right)</math> and <math>\Psi(w, cv) = c\Psi(w, v)</math> for {{nowrap|''c'' ∈ '''F'''<sub>''q''<sup>2</sup></sub>}}.{{clarify|What is K?|date=June 2014}} Further, all non-degenerate Hermitian forms on a vector space over a finite field<!-- clear from context, but I think it fails infinite fields of pos char--> are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to
: <math>\Psi(w, v) = w^\alpha \cdot v = \sum_{i=1}^n w_i^q v_i</math>
where <math>w_i,v_i</math> represent the coordinates of {{nowrap|''w'', ''v'' ∈ ''V''}} in some particular '''F'''<sub>''q''<sup>2</sup></sub>-basis of the ''n''-dimensional space ''V'' {{harv|Grove|2002|loc=Thm. 10.3}}.

Thus one can define a (unique) unitary group of dimension ''n'' for the extension '''F'''<sub>''q''<sup>2</sup></sub>/'''F'''<sub>''q''</sub>, denoted either as {{nowrap|U(''n'', ''q'')}} or {{nowrap|U(''n'', ''q''<sup>2</sup>)}} depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the '''special unitary group''' and denoted {{nowrap|SU(''n'', ''q'')}} or {{nowrap|SU(''n'', ''q''<sup>2</sup>)}}. For convenience, this article will use the {{nowrap|U(''n'', ''q''<sup>2</sup>)}} convention.  The center of {{nowrap|U(''n'', ''q''<sup>2</sup>)}} has order {{nowrap|''q'' + 1}} and consists of the scalar matrices that are unitary, that is those matrices ''cI<sub>V</sub>'' with <math>c^{q+1} = 1</math>.  The center of the special unitary group has order {{nowrap|gcd(''n'', ''q'' + 1)}} and consists of those unitary scalars which also have order dividing ''n''.  The quotient of the unitary group by its center is called the '''[[projective unitary group]]''', {{nowrap|PU(''n'', ''q''<sup>2</sup>)}}, and the quotient of the special unitary group by its center is the '''[[projective special unitary group]]''' {{nowrap|PSU(''n'', ''q''<sup>2</sup>)}}. In most cases ({{nowrap|''n'' > 1}} and {{nowrap|(''n'', ''q''<sup>2</sup>) ∉ {(2, 2<sup>2</sup>), (2, 3<sup>2</sup>), (3, 2<sup>2</sup>)}{{void}}}}), {{nowrap|SU(''n'', ''q''<sup>2</sup>)}} is a [[perfect group]] and {{nowrap|PSU(''n'', ''q''<sup>2</sup>)}} is a finite [[simple group]], {{harv|Grove|2002|loc=Thm. 11.22 and 11.26}}.