Although there may be some question of whether the implicit or the explicit form of one-to-one and invertible mappings should be called *Möbius transformations*, that is the name given to this class of transformations. Mappings often form a group. Since the associative law for their composition is a given, the items to be checked are closure, invertibility and the existence of the identity.

To check closure, consider the composite of two maps. First, suppose we have

t |
= | (29) | |

w |
= | (30) |

which gives, on substituting,

t |
= | (31) |

and, on simplifying,

t |
= | (32) |

which mimics the matrix product

From this it is apparent that the unimodular matrices form a representation of the group of Möbius transformations; in particular the identity is represented by the unit matrix,

z |
= | (33) |

and the inverse transformation by the inverse matrix.