A group has the presentation $\langle S \mid R \rangle$ if it is isomorphic to the quotient of the free group on $S$ by the normal subgroup generated by $R$. Every group has a presentation.

Examples

• A cyclic group of order $n$ has presentation $\langle x \mid x^n \rangle\,.$

• A free group on two generators has a simple presentation as $\langle x, y\rangle\,.$ If you want to consider a free abelian group on two generators, you need to impose the commutativity relations. So a free abelian group has presentation $\langle x,y \mid xyx^{-1}y^{-1}\rangle\,.$

• The dihedral group $D_n$ of order $2n$ has presentation $\langle r,s \mid s^2, r^n, srsr\rangle\,.$

Problems

Here are a few problems regarding group presentations.

• Group Isomorphism Problem — Given two groups presented in terms of generators and relations how can you tell if they are isomorphic?

• Word Problem for Groups — Given a group presented in terms of generators and relations and two words in the generators of the group, how can you tell if those words represent the same element?

These problems are difficult to answer. The first problem is generally unsolvable by a result of Adian and Rabin. And for the second, it's been proven that there's no algorithm that works for any group (see the Boone-Rogers theorem). Moral of the story being that a group presented in terms of generators and relations may be difficult to actually work with.