TITLE: What can be said about a group from its presentation?
QUESTION [12 upvotes]: This maybe a very general question.
If we have a group given by its presentation only, what kind of properties could be proven about it?
I know examples about non-amenability of some Burnside groups.
What kind of examples are there in literature where one proves some property "just" from a presentation?

REPLY [5 votes]: Given a group $\Gamma$ which you are interested in, given by a presentation, and an easy group $G$ which you understand (a dihedral group or a symmetric group, maybe), you can often use the presentation to determine whether or not there exists a surjective homomorphism $\Gamma\to G$, and maybe you can even count them. This is a strong tool for showing that two given presentations give rise to different groups, and for showing that your group must be "at least as complicated" as $G$. For instance, Tietze originally proved in 1908 that the trefoil is knotted by exhibiting a surjection from a Wirtinger presentation of the fundamental group of its complement onto the symmetric group $S_3$, while the fundamental group of the unknot is abelian and so can admit no such homomorphism.