Is coset enumeration the best way to get all the elements of a finitely presented group? if not what is the best algorithm to do this ? what difficulties could be caused by this process ( I mean storage difficulties ) ?

  • $\begingroup$ What do you mean by 'to get all the elements'? The group might be infinite. $\endgroup$ – Tara B Feb 22 '13 at 19:53
  • $\begingroup$ sorry i mean if it is known to be finite $\endgroup$ – J Nid Feb 22 '13 at 19:57
  • $\begingroup$ Do you mean by the coset enumeration as Todd-Coxeter Algorithm? $\endgroup$ – mrs Feb 22 '13 at 20:27
  • $\begingroup$ yes Todd-Coxeter $\endgroup$ – J Nid Feb 22 '13 at 20:29
  • 3
    $\begingroup$ The only genuine alternative to coset enumeration is the Knuth-Bendix completion algorithm. Coset enumeration (possibly combined with Reidemeister-Schreier, as suggested by Babak S) is usually faster on straightforward presentations of finite groups that are not too large. The Knuth-Bendix method computes a collection of reduction equations that enable you to reduce words in the group generators to a normal form. $\endgroup$ – Derek Holt Feb 23 '13 at 10:19

Personally, I have done many examples of finitely presented groups by TC algorithm. I like this method, however, sometimes it takes time, papers and its prime format. You might find some of them in my question, but there is another way which I am not familiar to it as I am with TC. It is called RSRP Reidemeister-Schreier Rewriting Process. Surely, for the proper explanation, we need a blackboard ;-) but what can I say that:

  • This method is suggested for finding the presentation of a subgroup with preferably small index.
  • It makes additional relations for the subgroup so it helps us to simplify the presentation more effective than TC.
  • But it makes additional generators which sometimes they are not depend on our original generators and this is bad.

Anyway, in the cases we face to the group with heavily relations and maybe long, this method is suggested and I have seen that.

  • $\begingroup$ I'm learning from you! +1 $\endgroup$ – Namaste Feb 23 '13 at 0:03

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