2

Due to the undecidability of some presentation related problems for groups, it may be hard to find an elegant presentation for a group, e.g. to find a finite presentation $\langle S\mid R\rangle$ for a finite group $G$, which is not the same as the non-freest group that satisfies $\langle S\mid R\rangle$ (there is other group satisfying $R$ but not equal to $G$). That said, finding a presentation might be more of an art than a science. Sadly, I haven't seen much of this art.

Can you share some beatiful (in your or many's eyes) techniques to find a presentation of a group? If you don't know of such technique, it would also be great if you can share some examples of finding a presentation cleverly.

Postscripts: for an elegant presentation, I hope the cardinality of the generators set $S$ is as small as possible (or at least not unnecessarily large, i.e., there is an improvement in $R$ compared with the smallest case), and the relations set $R$ doesn't contain redundant stuff (if there are some, as least as possible).

MJD
  • 67,568
  • 43
  • 308
  • 617
Raiden
  • 79
  • 3
    A clever (and easy) way is to ask GAP first. Or see this post for other methods (among others the answer by Derek Holt, mentioning the Todd-Coxeter coset enumeration). – Dietrich Burde Aug 28 '22 at 13:05
  • 2
    I am afraid that presentations found by computer algorithms might not satisfy the OP's idea of being beautiful or elegant. But I think the OP needs to be more precise about what qualities such a presentation should have. Otherwise answers will be a matter of opinion. – Derek Holt Aug 28 '22 at 13:13
  • 1
    I didn't say that. The primary purpose of Todd-Coxeter is to analyse given presentations, not to find presentations of given groups. When it is used as a tool to assist in finding presentations, the results typically have lots of redundant relations, and the relations found will just look like random words. – Derek Holt Aug 28 '22 at 13:17
  • 1
    OK, perhaps I misread the question. I see that it is primarily asking for clever techniques, but there is a suggestion that elegant results would be desirable. I'll withdraw my vote to close! – Derek Holt Aug 28 '22 at 13:19
  • 1
    I would also remark that finding a presentation of a finite group can be done algorithmically, but for infinite groups, such as infinite linear groups, human ingenuity is required. – Derek Holt Aug 28 '22 at 13:21
  • 1
    Re: postscript in the question. Sometimes the generating set of the smallest size might not be the most "elegant" one. For example, finite simple groups are generated by two elements. But for those, I suppose the relations are not so pretty. You could argue that the presentation given by Steinberg relations (with many more generators) is elegant and nice, or at least understandable by a human. – spin Aug 28 '22 at 13:44
  • The only technique I've seen is a reduction of powers strategy where you try to find some elements so that $x^k = y^j$ with $j < k$ through guess and test. – CyclotomicField Aug 28 '22 at 14:45
  • 2
    @spin The clearest example of this is the symmetric group. Iwould argue that the Coxeter presentation is the most elegant, even though a presentation with the generators $(1,2)$ and $(1,2,\dots,n)$ actually has (much) shorter length! – David A. Craven Aug 28 '22 at 14:54

0 Answers0