Projective special linear group:PSL(2,R)

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Definition

Definition as a matrix group

$PSL(2,\mathbb {R} )$ is defined as the projective special linear group of degree two over the field of real numbers.

In other words, it is defined as $SL(2,\mathbb {R} )/\{\pm I\}$.

Definition as a group of fractional linear transformations

The group $PSL(2,\mathbb {R} )$ can be defined as the group of conformal automorphisms of the upper half-plane. In other words, it is the group of maps of the form:

$z\mapsto {\frac {az+b}{cz+d}}$.

where $a,b,c,d\in \mathbb {C}$ are such that these maps send the upper half-plane to itself.

Arithmetic functions

Function	Value	Similar groups	Explanation
order of a group	cardinality of the continuum		Same infinite cardinality as its double cover SL(2,R).
exponent of a group	infinite		Same as its double cover SL(2,R).
composition length	1	groups with same composition length	Follows from being simple -- see projective special linear group is simple.
chief length	1	groups with same chief length	Follows from being simple -- see projective special linear group is simple.
dimension of an algebraic group	3	groups with same dimension of an algebraic group	As $PSL(n,\_),n=2:n^{2}-1=2^{2}-1=3$
dimension of a real Lie group	3	groups with same dimension of a real Lie group	As $PSL(n,\mathbb {R} ),n=2:n^{2}-1=2^{2}-1=3$