General linear group:GL(2,R)

Definition

The general linear group $GL(2,\mathbb {R} )$ is defined as the group of invertible $2\times 2$ matrices with entries from the field of real numbers, and with the group operation being matrix multiplication.

It is a particular case of general linear group of degree two, general linear group over reals, and, more generally, of a general linear group.

Structures

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.

Arithmetic functions

Function	Value	Similar groups	Explanation
dimension of an algebraic group	4		As $GL(n,\mathbb {R} ),n=2$: $n^{2}=2^{2}=4$
dimension of a real Lie group	4		As $GL(n,\mathbb {R} ),n=2$: $n^{2}=2^{2}=4$

Group properties

Abstract group properties

Property	Satisfied?	Explanation
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 $SL(2,\mathbb {R} )$.
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 $\mathbb {R} ^{4}$.

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.