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# 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

$G=\langle g_{i}\mid r_{j}\rangle,$ |

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 $1,\ldots,n$ admits the following finite presentation (Note: this presentation is not canonical. Other presentations are known.) As generators take

$g_{i}=(i,i+1),\quad i=1,\ldots,n-1,$ |

the transpositions of adjacent elements. As defining relations take

$(g_{i}g_{j})^{{n_{{i,j}}}}=\mathrm{id},\quad i,j=1,\ldots n,$ |

where

$\displaystyle n_{{i,i}}$ | $\displaystyle=1$ | ||

$\displaystyle n_{{i,i+1}}$ | $\displaystyle=3$ | ||

$\displaystyle n_{{i,j}}$ | $\displaystyle=2,\quad|j-i|>1.$ |

This means that a finite symmetric group is a Coxeter group.

## Mathematics Subject Classification

20A05*no label found*20F05

*no label found*

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