# Projective special linear group:PSL(2,R)

From Groupprops

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## Contents

## Definition

### Definition as a matrix group

is defined as the projective special linear group of degree two over the field of real numbers.

In other words, it is defined as .

### Definition as a group of fractional linear transformations

The group 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:

.

where 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 |

dimension of a real Lie group | 3 | groups with same dimension of a real Lie group | As |