General Linear Group

The general linear group over a ring with identity is the group of all invertible matrices with entries in . When is a field , it can be written as

(1)

where denotes the set of all matrices over . Equivalently, is the group of invertible linear transformations of the vector space .

The determinant is a surjective group homomorphism from to the multiplicative group . Its kernel is the special linear group, giving the exact sequence

(2)

The group center consists of the nonzero scalar matrices , where is the identity matrix. Factoring by this group center gives the projective general linear group.

For the finite field , the group order can be found by choosing successively linearly independent column vectors. Writing for the group order of a finite group , the result is

(3)

Over the real numbers, is a real Lie group of dimension with two components, distinguished by the sign of the determinant. The complex Lie group is connected and has complex dimension .