General linear group over reals

Definition

The general linear group over reals of degree $n$, denoted $GL(n,\R)$ or $GL_n(\R)$, is defined as the general linear group of degree $n$ over the field of real numbers $\R$.

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 $GL(n,\R)$ 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	$n^2$		The group is a Zariski open subset of the matrix algebra (which can be identified with $\R^{n^2}$), defined as the set of elements of nonzero determinant. Note that the determinant map is a polynomial map.
dimension of a real Lie group	$n^2$		The group is an open subset of the matrix algebra (which can be identified with $\R^{n^2}$), defined as the set of elements of nonzero determinant. Note that the determinant map is jointly continuous

Group properties

Abstract group properties

Property	Satisfied?	Explanation
abelian group	No except for $n = 1$	Follows from center of general linear group is group of scalar matrices over center
nilpotent group	No except for $n = 1$	Follows from special linear group is quasisimple
solvable group	No except for $n = 1$	Follows from special linear group is quasisimple
simple group	No	Has a proper nontrivial center, also has normal subgroup $SL(2,\R)$.
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 $\R^4$.

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.