Presentation

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. [a1], Sect. 1.2. Given a presentation of a group, there are systematic ways for obtaining presentations of subgroups and quotient groups.

References

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