General linear group over reals
Some of the properties of these general linear groups generalize to general linear groups over fields that resemble the reals in one or more important respect: for instance, formally real fields, totally real fields, ordered fields, Pythagorean fields, and quadratically closed fields.
Each group can be thought of in any of the following ways:
- It is a real Lie group.
- It is a linear algebraic group over the field of real numbers (note that this is not an algebraically closed field).
- It is a topological group.
|dimension of an algebraic group||The group is a Zariski open subset of the matrix algebra (which can be identified with ), defined as the set of elements of nonzero determinant. Note that the determinant map is a polynomial map.|
|dimension of a real Lie group||The group is an open subset of the matrix algebra (which can be identified with ), defined as the set of elements of nonzero determinant. Note that the determinant map is jointly continuous|
Abstract group properties
|abelian group||No except for||Follows from center of general linear group is group of scalar matrices over center|
|nilpotent group||No except for||Follows from special linear group is quasisimple|
|solvable group||No except for||Follows from special linear group is quasisimple|
|simple group||No||Has a proper nontrivial center, also has normal subgroup .|
|almost simple group||No||Has a nontrivial center.|
|quasisimple group||No||Not a perfect group; has a nontrivial homomorphism to an abelian group, namely the determinant map|
|almost quasisimple group||Yes||Follows from special linear group is quasisimple|
Topological/Lie group properties
The topology here is the subspace topology from the Euclidean topology on the set of all matrices, which is identified with the Euclidean space .
|connected topological group||No||the matrices with positive determinant form one connected component. The matrices with negative determinant form the other connected component.|
|compact group||No||The determinant map is continuous and surjective to the non-compact set of nonzero reals.|