General linear group:GL(2,R)
The general linear group is defined as the group of invertible matrices with entries from the field of real numbers, and with the group operation being matrix multiplication.
The 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||4||As :|
|dimension of a real Lie group||4||As :|
Abstract group properties
|abelian group||No||Follows from center of general linear group is group of scalar matrices over center|
|nilpotent group||No||Follows from special linear group is quasisimple|
|solvable group||No||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.|