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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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One easy way (that is, easy if it works) to determine if $G$ is infinite is to find a homomorphism out of $G$ onto some other group that you know to be infinite. For example, it is straightforward to determine if the abelianization of a finitely presented group is infinite (we get a finitely presented abelian group and we can compute Smith normal forms to determine its rank). In this case the abelianization is trivial so that doesn't work. –  Qiaochu Yuan Sep 10 '12 at 1:37
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I changed $G=< u, v~|~~ uv^2=v^3u, ~u^2v=vu^3>$ to $G=\langle u, v\mid uv^2=v^3u, ~u^2v=vu^3\rangle$. I used \langle, \rangle, and \mid. Notice the spacing that results from the use of \mid. This spacing need not be added manually. –  Michael Hardy Sep 10 '12 at 1:39
    
@QiaochuYuan Do you mean a homomorphism out of $G$ onto some infinite group? The trivial homomorphism into, say, $\mathbb{C}$ is certainly a homomorphism from any group into an infinite group ... –  Neal Sep 10 '12 at 1:39
    
@Neal: yes, thanks for the correction. –  Qiaochu Yuan Sep 10 '12 at 1:40
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Here is a proof your group (and many like it) are trivial: math.stackexchange.com/questions/66573/… –  user641 Sep 10 '12 at 1:42

1 Answer 1

up vote 5 down vote accepted

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);

1

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thanks, I have not used GAP before. –  ougao Sep 10 '12 at 1:56
    
Sure. No problem. GAP is free and it's a fantastic tool. –  Bill Cook Sep 10 '12 at 2:05
    
If it's a homework question, then doing it using GAP is cheating! (Although I have to admit that I would be tempted to do that first just to get an idea of what the answer might be, before trying to prove it by hand.) –  Derek Holt Sep 10 '12 at 8:00
    
@Derek, good suggestion, thanks! –  ougao Sep 11 '12 at 2:58

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