General linear group over reals
From Groupprops
Contents
Definition
The general linear group over reals of degree , denoted
or
, is defined as the general linear group of degree
over the field of real numbers
.
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.
Structures
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.
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
dimension of an algebraic group | ![]() |
The group is a Zariski open subset of the matrix algebra (which can be identified with ![]() | |
dimension of a real Lie group | ![]() |
The group is an open subset of the matrix algebra (which can be identified with ![]() |
Group properties
Abstract group properties
Property | Satisfied? | Explanation |
---|---|---|
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 .
Property | Satisfied? | Explanation |
---|---|---|
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. |