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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.

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    You should notice that your presentation is "balanced". That is, there is an equal number of generators and relations. However, if you have just one more generator than relations then your group is automatically infinite, while if you have at least two more generators then relations then your group is what is called "Large". That is, it has a subgroup of finite index which maps onto $F_2$, the free group on two generators. Largeness might seem synthetic, but it is a natural and rather interesting property.2012-09-10

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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...

f := FreeGroup("u","v");;

u := f.1;; v := f.2;;

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 ]

G := f/rels;

Size(G);

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    @Derek, good suggestion, thanks!2012-09-11