Find the order of a group G from its presentation

Question

Suppose G is a group defined by the presentation $G=\langle u, v\mid uv^2=v^3u, ~u^2v=vu^3\rangle$, is $G$ finite or infinite? If it is finite, what is its order?

In general, I want to know whether there are some useful strageties to detect whether a group $G$ is finite or infinite only from its presentation, and when it is finite, how to deduce its order.

Answer 1

One method for attempting to determine the order of a group given a finite presentation is the Todd-Coxeter algorithm. Be warned, it's not fun to try to run through by hand.

One can prove that there is no algorithm which can detect whether finite presentations yield finite or infinite groups. In fact, the problem of merely detecting whether you have the trivial group or not is unsolvable.

For your specific problem GAP says your group is trivial...

gap> f := FreeGroup("u","v");;
gap> u := f.1;; v := f.2;;
gap> rels := [ u*v^2*u^(-1)*v^(-3), u^2*v*u^(-3)*v^(-1) ];
[ u*v^2*u^-1*v^-3, u^2*v*u^-3*v^-1 ]
gap> G := f/rels;
<fp group on the generators [ u, v ]>
gap> Size(G);
1