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