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

From Groupprops

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## Definition

The group , also sometimes called the **modular group**, is defined in the following equivalent ways:

- It is the projective special linear group of degree two over the ring of integers. In other words, it is the quotient of the special linear group:SL(2,Z) by the subgroup .
- It is the inner automorphism group of braid group:B3, i.e., the quotient of by its center.
- It is the free product of cyclic group:Z2 and cyclic group:Z3.

### Definition by presentation

The group can be defined by the following presentation, where denotes the identity element:

- As a projective special linear group: where is the image of the matrix and is the image of the matrix .
- As a matrix group:

Function | Value | Explanation |
---|---|---|

order | infinite (countable) | As : The group is the quotient of the countably infinite group by its center, which is a subgroup of order two. As a free product: any free product of nontrivial finite groups is countably infinite. |

exponent | infinite | As : The group is the quotient of , which has infinite exponent, by a finite subgroup. More explicitly, the image of has infinite order. |

minimum size of generating set | 2 | As : Follows from being 2-generated. As inner automorphism group of braid group : Follows from being 2-generated. As free product of cyclic group:Z2 and cyclic group:Z3: Follows because each of the free factors is cyclic, so we get a generating set of size 2. NOTE: In all interpretations, we can rule out the possibility of a generating set of size 1 because cyclic implies abelian and the group is non-abelian. |

subgroup rank | infinite (countable) | Use that has infinite subgroup rank |

## Group properties

Property | Satisfied? | Explanation | Corollary properties satisfied/dissatisfied |
---|---|---|---|

2-generated group | Yes | See explanation for minimum size of generating set above | satisfies: finitely generated group, countable group |

Noetherian group | No | See explanation for subgroup rank above | |

finitely presented group | Yes | Any of the definitions (, free product) gives a finite presentation | |

solvable group | No | contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two | dissatisfies: nilpotent group, abelian group |

group satisfying no nontrivial identity | Yes | contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two and note that the image in of the Sanov subgroup is isomorphic to it. | |

SQ-universal group | Yes | contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two and note that the image in of the Sanov subgroup is isomorphic to it. | |

residually finite group | Yes | The kernels of the homomorphisms for natural numbers are normal subgroups of finite index and their intersection is trivial. | satisfies: finitely generated residually finite group |

Hopfian group | Yes | Follows from finitely generated and residually finite implies Hopfian | satisfies: finitely generated Hopfian group |

## GAP implementation

Description | Functions used |
---|---|

FreeProduct(CyclicGroup(2),CyclicGroup(3)) |
FreeProduct, CyclicGroup |