Given a field , the general linear group (or ) is the group of invertible linear transformations of the vector space . It can be canonically identified with the group of matrices with entries in having nonzero determinant.
This group can be considered as a (quasi-affine) subvariety of the affine space of square matrices of size defined by the condition that the determinant of a matrix is nonzero. It can be also presented as an affine subvariety of the affine space defined by the equation (where varies over the factor and over the factor ).
One can in fact consider the set of invertible matrices over an arbitrary unital ring, not necessarily commutative. Thus becomes a presheaf of groups on where one can take rings either in commutative or in noncommutative sense. In the commutative case, this functor defines a group scheme; it is in fact the affine group scheme represented by the commutative ring .
The above is sometimes referred to as the unstable general linear group, whilst the result if one lets go to infinity is called the stable general linear group of , and is then written with no suffix.
O.T. O’Meara, Lectures on Linear Groups, Amer. Math. Soc., Providence, RI, 1974.
B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.