General Unitary Group

The general unitary group is the group of invertible linear transformations of that preserve a nonsingular Hermitian form. All such Hermitian forms of a fixed dimension over are equivalent. In a basis in which the Hermitian form has matrix ,

$GU_n(q)={A in GL_n(q^2):A^_^THA=H},$

(1)

where the bar denotes entrywise conjugation by the field automorphism . This is the analogue over a finite field of the unitary group.

For a finite group , denotes its group order. In particular,

(2)

The determinant of every element has norm 1 in . The determinant group homomorphism is onto the cyclic group of elements having norm 1, whose group order is , and its kernel is the special unitary group. Thus

$1->SU_n(q)->GU_n(q)->^(det){a in F_(q^2)^×:a^(q+1)=1}->1.$

(3)

Some authors denote this full group of isometries by ; is the convention of the Atlas of Finite Groups.

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups

, and

." §2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985.

Grove, L. C. Classical Groups and Geometric Algebra. Providence, RI: American Mathematical Society, 2002.

See also