# Special linear group:SL(2,C)

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

The group is defined as the group of matrices with entries from the field of complex numbers and determinant , under matrix multiplication.

.

It is a particular case of a special linear group over complex numbers, special linear group of degree two, and hence of a special linear group.

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order of a group | cardinality of the continuum | The cardinality is at least that of the continuum, because we can inject into this group by . On the other hand, it is a subset of , so the cardinality is not more than that of the continuum. | |

exponent of a group | infinite | there exist elements, such as , of infinite order. | |

composition length | 2 | groups with same composition length | Center is simple (isomorphic to cyclic group:Z2) and the quotient group PSL(2,C) is also simple. |

chief length | 2 | groups with same chief length | Similar reason to composition length. |

dimension of an algebraic group | 3 | groups with same dimension of an algebraic group | As |

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

dimension of a real Lie group | 6 | groups with same dimension of a real Lie group | Twice its dimension as a complex Lie group. |

## Group properties

### Abstract group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

quasisimple group | Yes | special linear group is quasisimple (with a couple of finite exceptions). Its inner automorphism group, which is projective special linear group:PSL(2,C), is simple. |

simple non-abelian group | No | The center is , so is proper and nontrivial. |

### Topological/Lie group properties

Property | Satisfied? | Explanation |
---|---|---|

connected topological group | Yes | It is generated by matrices of the form and . Both sets are connected sets are matrices containing the identity, so the group is connected. |

compact group | No | It contains matrices of the form where the can be arbitrarily large, so is not compact as a subset of . |

## Elements

`Further information: element structure of special linear group:SL(2,C)`

## Linear representation theory

`Further information: linear representation theory of special linear group:SL(2,C)`