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I wrote a computer program that generates group presentations.

I would like to know the sizes of the resulting groups. I know that this is undecidable.

Are there good heuristic programs that can try to compute the size of a group given by generators and relations?

I am not interested in apporximating the size. Only in determining the exact size, when the software can do it, hopefully often in my cases.

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You can use Magma Online Calculator. For instance, the following code:

 F<a, b> := FreeGroup(2);  

 G<x, y>, phi := quo< F | a^2, b^3, (a*b)^4 >;  

 #G
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  • $\begingroup$ It gives me: "User error: Identifier 'a' has not been declared or assigned" $\endgroup$ – Alex Jun 12 '13 at 13:17
  • $\begingroup$ @O.L.: Would you like me to edit? I edit and if it is as you wanted keep it. ;) $\endgroup$ – mrs Jun 12 '13 at 13:27
  • $\begingroup$ @O.L.: I think Daniel did it right. But you may want F<a,b> as $F\langle a,b\rangle$. If so, you can do it via F\langle a,b\rangle between two $. $\endgroup$ – mrs Jun 12 '13 at 13:31
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    $\begingroup$ @BabakS. No, now it's fine, thanks! I still have some problems with Mathjax. I tried to look at the code of your answer and edit by Daniel Rust, there are no special symbols used, but when I type this myself, it doesn't look the same (line breaks, problems with ><, etc). +1 by the way. $\endgroup$ – Start wearing purple Jun 12 '13 at 13:35
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You can use GAP as well. Regarding to what O.L gave you I am posting an example accordingly:

 gap> f:=FreeGroup("a","b");;
 gap> a:=f.1;;    b:=f.2;;
 gap> g:=f/[a^2,b^3,(a*b)^4];
 gap> Elements(g);;
 gap> Size(g)

If you be familiar to use this software, then you'll wish to use it in sleep even!! It is indeed a wonderful and of course a powerful tool.

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  • $\begingroup$ Yay for GAP!! ;-) $\endgroup$ – Namaste Jun 13 '13 at 0:11
  • $\begingroup$ and GAP is included in sage: sagemath.org $\endgroup$ – kjetil b halvorsen Jun 13 '13 at 10:38

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