# presentation of a group

A *presentation ^{}* of a group $G$ is a description of $G$ in terms of
generators and relations (sometimes also known as relators).
We say that the group is finitely
presented, if it can be described in terms of a finite number of
generators

^{}and a finite number of defining relations. A collection

^{}of group elements ${g}_{i}\in G,i\in I$ is said to generate $G$ if every element of $G$ can be specified as a product

^{}of the ${g}_{i}$, and of their inverses

^{}. A relation

^{}is a word over the alphabet consisting of the generators ${g}_{i}$ and their inverses, with the property that it multiplies out to the identity

^{}in $G$. A set of relations ${r}_{j},j\in J$ is said to be defining, if all relations in $G$ can be given as a product of the ${r}_{j}$, their inverses, and the $G$-conjugates of these.

The standard notation for the presentation of a group is

$$G=\u27e8{g}_{i}\mid {r}_{j}\u27e9,$$ |

meaning that $G$ is generated by generators ${g}_{i}$, subject to
relations ${r}_{j}$. Equivalently, one has a short exact sequence^{} of
groups

$$1\to N\to F[I]\to G\to 1,$$ |

where $F[I]$ denotes the free group^{}
generated by the ${g}_{i}$, and where $N$ is the smallest normal subgroup
containing all the ${r}_{j}$. By the Nielsen-Schreier Theorem, the kernel $N$
is itself a free group, and hence we assume without loss of generality
that there are no relations among the relations.

Example. The symmetric group^{} on $n$ elements $1,\mathrm{\dots},n$
admits the following finite presentation (Note: this presentation is
not canonical. Other presentations are known.) As generators take

$${g}_{i}=(i,i+1),i=1,\mathrm{\dots},n-1,$$ |

the transpositions^{} of adjacent elements. As defining relations take

$${({g}_{i}{g}_{j})}^{{n}_{i,j}}=\mathrm{id},i,j=1,\mathrm{\dots}n,$$ |

where

${n}_{i,i}$ | $=1$ | ||

${n}_{i,i+1}$ | $=3$ | ||

${n}_{i,j}$ | $=2,|j-i|>1.$ |

This means that a finite symmetric group is a Coxeter group^{}.

Title | presentation of a group |

Canonical name | PresentationOfAGroup |

Date of creation | 2013-03-22 12:23:23 |

Last modified on | 2013-03-22 12:23:23 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 20 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 20A05 |

Classification | msc 20F05 |

Synonym | presentation |

Synonym | finite presentation |

Synonym | finitely presented |

Related topic | GeneratingSetOfAGroup |

Related topic | CayleyGraph |

Defines | generator |

Defines | relation |

Defines | generators and relations |

Defines | relator |