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''of a group''

A specification of a [[group]] by generators and relations among them.

====Comments====
Every group can be presented by means of generators and relations. A presentation is finitely generated, respectively finitely related, if the number of generators, respectively relations, is finite. A finite presentation is one with both a finite number of relations and a finite number of generators. A presentation of the [[symmetric group]] $S_n$ of permutations on $n$ letters is as follows: there are $n-1$  generators $\sigma_2,\ldots,\sigma_n$, and the relations are $\sigma_i^2 = e$, $\sigma_i\sigma_j = \sigma_j\sigma_i$ if $|i-j| \ge 2$, $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$. If the relations $\sigma_i^2 = e$ are removed, one obtains a presentation of the [[braid group]] $B_n$.

If $G$ is presented by generators $G_i,$, $i \in I$, and relations $R_j$, $j \in J$, one writes $G = \langle G_i | R_j \rangle$. In that case $G$ is the quotient group of the [[free group]] on the generators $G_i,$ by the normal subgroup generated by the relations $R_j$. For details cf. [[#References|[a1]]], Sect. 1.2. Given a presentation of a group, there are systematic ways for obtaining presentations of subgroups and quotient groups.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations of groups in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   W.O.J. Moser,   "Generators and relations for discrete groups" , Springer  (1965)</TD></TR>
</table>

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