General linear group:GL(2,R)

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Definition

The general linear group GL(2,\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:

Arithmetic functions

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