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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.

Shaun
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ougao
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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 01:37
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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 01:39
  • @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 01:39
  • @Neal: yes, thanks for the correction. – Qiaochu Yuan Sep 10 '12 at 01:40
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    Here is a proof your group (and many like it) are trivial: http://math.stackexchange.com/questions/66573/a-particular-two-variable-system-in-a-group/66632#66632 –  Sep 10 '12 at 01:42
  • @ Steve, many thanks! – ougao Sep 10 '12 at 01:52
  • @ougao: The link Steve gave you is the best answer. Still, you can find more than you had expected by your question by following the link. I think the best source for such these groups is Presentations of groups by D. L. Johnson. Todd-coxeter Algorithm and Tietze transformations are ways which help you step by step finding the order of such this group. :) – Mikasa Sep 10 '12 at 07:29
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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. – user1729 Sep 10 '12 at 10:18
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    As I recall even determining from a presentation whether a group is trivial is an NP-hard problem. – Travis Willse Aug 04 '23 at 23:34

1 Answers1

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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...

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);
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Olexandr Konovalov
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Bill Cook
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