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.


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 >;  

  • $\begingroup$ 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
  • 1
    $\begingroup$ @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

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.

  • $\begingroup$ 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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